Coefficient a

The first nine terms of geometric series 1 + r + r² + r³ ... drawn as functions (colored in the order red, green, blue, red, green, blue, ...) within the range |r| < 1. The closed form geometric series 1 / (1 - r) is the black dashed line.

The geometric series a + ar + ar² + ar³ + ... is written in expanded form.^[1] Every coefficient in the geometric series is the same. In contrast, the power series written as a₀ + a₁r + a₂r² + a₃r³ + ... in expanded form has coefficients a_i that can vary from term to term. In other words, the geometric series is a special case of the power series. The first term of a geometric series in expanded form is the coefficient a of that geometric series.

In addition to the expanded form of the geometric series, there is a generator form^[1] of the geometric series written as

${\displaystyle \sum _{k=0}^{\infty }ar^{k}}$

and a closed form of the geometric series written as

${\displaystyle {\frac {a}{1-r}}{\text{ for }}|r|<1.}$

The derivation of the closed form from the expanded form is shown in this article's § Sum section. However even without that derivation, the result can be confirmed with long division: a divided by (1 - r) results in a + ar + ar² + ar³ + ... , which is the expanded form of the geometric series.

It is often a convenience in notation to set the series equal to the sum s and work with the geometric series

s = a + ar + ar² + ar³ + ar⁴ + ... in its normalized form

s / a = 1 + r + r² + r³ + r⁴ + ... or in its normalized vector form

s / a = [1 1 1 1 1 ...][1 r r² r³ r⁴ ...]^T or in its normalized partial series form

s_n / a = 1 + r + r² + r³ + r⁴ + ... + rⁿ, where n is the power (or degree) of the last term included in the partial sum s_n.

Changing even one of the coefficients to something other than coefficient a would change the resulting sum of functions to some function other than a / (1 − r) within the range |r| < 1. As an aside, a particularly useful change to the coefficients is defined by the Taylor series, which describes how to change the coefficients so that the sum of functions converges to any user selected, sufficiently smooth function within a range.

Common ratio r

The convergence of the geometric series with r=1/2 and a=1/2

The convergence of the geometric series with r=1/2 and a=1

Close-up view of the cumulative sum of functions within the range -1 < r < -0.5 as the first 11 terms of the geometric series 1 + r + r² + r³ + ... are added. The geometric series 1 / (1 - r) is the red dashed line.

The geometric series a + ar + ar² + ar³ + ... is an infinite series defined by just two parameters: coefficient a and common ratio r. Common ratio r is the ratio of any term with the previous term in the series. Or equivalently, common ratio r is the term multiplier used to calculate the next term in the series. The following table shows several geometric series:

The convergence of the geometric series depends on the value of the common ratio r:

• If |r| < 1, the terms of the series approach zero in the limit (becoming smaller and smaller in magnitude), and the series converges to the sum a / (1 - r).
• If |r| = 1, the series does not converge. When r = 1, all of the terms of the series are the same and the series is infinite. When r = −1, the terms take two values alternately (for example, 2, −2, 2, −2, 2,... ). The sum of the terms oscillates between two values (for example, 2, 0, 2, 0, 2,... ). This is a different type of divergence. See for example Grandi's series: 1 − 1 + 1 − 1 + ···.
• If |r| > 1, the terms of the series become larger and larger in magnitude. The sum of the terms also gets larger and larger, and the series does not converge to a sum. (The series diverges.)

The rate of convergence also depends on the value of the common ratio r. Specifically, the rate of convergence gets slower as r approaches 1 or −1. For example, the geometric series with a = 1 is 1 + r + r² + r³ + ... and converges to 1 / (1 - r) when |r| < 1. However, the number of terms needed to converge approaches infinity as r approaches 1 because a / (1 - r) approaches infinity and each term of the series is less than or equal to one. In contrast, as r approaches −1 the sum of the first several terms of the geometric series starts to converge to 1/2 but slightly flips up or down depending on whether the most recently added term has a power of r that is even or odd. That flipping behavior near r = −1 is illustrated in the adjacent image showing the first 11 terms of the geometric series with a = 1 and |r| < 1.

The common ratio r and the coefficient a also define the geometric progression, which is a list of the terms of the geometric series but without the additions. Therefore the geometric series a + ar + ar² + ar³ + ... has the geometric progression (also called the geometric sequence) a, ar, ar², ar³, ... The geometric progression - as simple as it is - models a surprising number of natural phenomena,

• from some of the largest observations such as the expansion of the universe where the common ratio r is defined by Hubble's constant,
• to some of the smallest observations such as the decay of radioactive carbon-14 atoms where the common ratio r is defined by the half-life of carbon-14.

As an aside, the common ratio r can be a complex number such as |r|e^iθ where |r| is the vector's magnitude (or length), θ is the vector's angle (or orientation) in the complex plane and i² = -1. With a common ratio |r|e^iθ, the expanded form of the geometric series is a + a|r|e^iθ + a|r|²e^i2θ + a|r|³e^i3θ + ... Modeling the angle θ as linearly increasing over time at the rate of some angular frequency ω₀ (in other words, making the substitution θ = ω₀t), the expanded form of the geometric series becomes a + a|r|e^iω₀t + a|r|²e^i2ω₀t + a|r|³e^i3ω₀t + ... , where the first term is a vector of length a not rotating at all, and all the other terms are vectors of different lengths rotating at harmonics of the fundamental angular frequency ω₀. The constraint |r|<1 is enough to coordinate this infinite number of vectors of different lengths all rotating at different speeds into tracing a circle, as shown in the adjacent video. Similar to how the Taylor series describes how to change the coefficients so the series converges to a user selected sufficiently smooth function within a range, the Fourier series describes how to change the coefficients (which can also be complex numbers in order to specify the initial angles of vectors) so the series converges to a user selected periodic function.

Proof of convergence

We can prove that the geometric series converges using the sum formula for a geometric progression:

$${\displaystyle {\begin{aligned}1+r+r^{2}+r^{3}+\cdots \ &=\lim _{n\to \infty }\left(1+r+r^{2}+\cdots +r^{n}\right)\\&=\lim _{n\to \infty }{\frac {1-r^{n+1}}{1-r}}.\end{aligned}}}$$

The second equality is true because if ${\displaystyle |r|<1,}$ then ${\displaystyle r^{n+1}\to 0}$ as ${\displaystyle n\to \infty }$ and

$${\displaystyle {\begin{aligned}\left(1+r+r^{2}+\cdots +r^{n}\right)(1-r)&=\left((1-r)+(r-r^{2})+(r^{2}-r^{3})+\dots +(r^{n}-r^{n+1})\right)\\&=1+(-r+r)+(-r^{2}+r^{2})+\dots +(-r^{n}+r^{n})-r^{n+1}\\&=1-r^{n+1}.\end{aligned}}}$$

Alternatively, a geometric interpretation of the convergence is shown in the adjacent diagram. The area of the white triangle is the series remainder = s − s_n = arⁿ⁺¹ / (1 − r). Each additional term in the partial series reduces the area of that white triangle remainder by the area of the trapezoid representing the added term. The trapezoid areas (i.e., the values of the terms) get progressively thinner and shorter and closer to the origin. In the limit, as the number of trapezoids approaches infinity, the white triangle remainder vanishes as it is filled by trapezoids and therefore s_n converges to s, provided |r|<1. In contrast, if |r|>1, the trapezoid areas representing the terms of the series instead get progressively wider and taller and farther from the origin, not converging to the origin and not converging as a series.

Rate of convergence

Converging alternating geometric series with common ratio r = -1/2 and coefficient a = 1. (TOP) Alternating positive and negative areas. (MIDDLE) Gaps caused by addition of adjacent areas. (BOTTOM) Gaps filled by broadening and decreasing the heights of the separated trapezoids.

After knowing that a series converges, there are some applications in which it is also important to know how quickly the series converges. For the geometric series, one convenient measure of the convergence rate is how much the previous series remainder decreases due to the last term of the partial series. Given that the last term is arⁿ and the previous series remainder is s - s_n-1 = arⁿ / (1 - r)), this measure of the convergence rate of the geometric series is arⁿ / (arⁿ / (1 - r)) = 1 - r, if 0 ≤ r < 1.

If r < 0, adjacent terms in the geometric series alternate between being positive and negative. A geometric interpretation of a converging alternating geometric series is shown in the adjacent diagram in which the areas of the negative terms are shown below the x axis. Pairing and summing each positive area with its negative smaller area neighbor results in non-overlapped trapezoids separated by gaps. To remove the gaps, broaden each trapezoid to cover the rightmost 1 - r² of the original triangle area instead of just the rightmost 1 - |r|. However, to maintain the same trapezoid areas during this broadening transformation, scaling is needed: scale*(1 - r²) = (1 - |r|), or scale = (1 - |r|) / (1 - r²) = (1 + r) / (1 - r²) = (1 + r) / ((1 + r)(1 - r)) = 1 / (1 - r) where -1 < r ≤ 0. Note that because r < 0 this scale decreases the amplitude of the separated trapezoids in order to fill in the separation gaps. In contrast, for the case r > 0 the same scale 1 / (1 - r) increases the amplitude of the non-overlapped trapezoids in order to account for the loss of the overlapped areas.

With the gaps removed, pairs of terms in a converging alternating geometric series become a converging (non-alternating) geometric series with common ratio r² to account for the pairing of terms, coefficient a = 1 / (1 - r) to account for the gap filling, and the degree (i.e., highest powered term) of the partial series called m instead of n to emphasize that terms have been paired. Similar to the r > 0 case, the r < 0 convergence rate = ar^2m / (s - s_m-1) = 1 - r², which is the same as the convergence rate of a non-alternating geometric series if its terms were similarly paired. Therefore, the convergence rate does not depend upon n or m and, perhaps more surprising, does not depend upon the sign of the common ratio. One perspective that helps explain the variable rate of convergence that is symmetric about r = 0 is that each added term of the partial series makes a finite contribution to the infinite sum at r = 1 and each added term of the partial series makes a finite contribution to the infinite slope at r = -1.

Derivation

Finite series

To derive this formula, first write a general geometric series as:

$${\displaystyle \sum _{k=1}^{n}ar^{k-1}=ar^{0}+ar^{1}+ar^{2}+ar^{3}+\cdots +ar^{n-1}.}$$

We can find a simpler formula for this sum by multiplying both sides of the above equation by 1 − r, and we'll see that

$${\displaystyle {\begin{aligned}(1-r)\sum _{k=1}^{n}ar^{k-1}&=(1-r)\left(ar^{0}+ar^{1}+ar^{2}+ar^{3}+\cdots +ar^{n-1}\right)\\&=ar^{0}+ar^{1}+ar^{2}+ar^{3}+\cdots +ar^{n-1}-ar^{1}-ar^{2}-ar^{3}-\cdots -ar^{n-1}-ar^{n}\\&=a-ar^{n}\end{aligned}}}$$

since all the other terms cancel. If r ≠ 1, we can rearrange the above to get the convenient formula for a geometric series that computes the sum of n terms:

$${\displaystyle \sum _{k=1}^{n}ar^{k-1}={\frac {a(1-r^{n})}{1-r}}.}$$

Related formulas

If one were to begin the sum not from k=1 or 0 but from a different value, say ${\displaystyle m}$, then

$${\displaystyle {\begin{aligned}\sum _{k=m}^{n}ar^{k-1}&={\begin{cases}{\frac {a(r^{m-1}-r^{n})}{1-r}}&{\text{if }}r\neq 1\\a(n-m+1)&{\text{if }}r=1\end{cases}}\\\sum _{k=m}^{n}ar^{k}&={\begin{cases}a(n-m+1)&{\text{if }}r=1\\{\frac {a(r^{m}-r^{n+1})}{1-r}}&{\text{if }}r\neq 1\end{cases}}\end{aligned}}}$$

Differentiating this formula with respect to ${\displaystyle r}$ allows us to arrive at formulae for sums of the form

$${\displaystyle G_{s}(n,r):=\sum _{k=0}^{n}k^{s}r^{k}.}$$

For example:

$${\displaystyle {\frac {d}{dr}}\sum _{k=0}^{n}r^{k}=\sum _{k=1}^{n}kr^{k-1}={\frac {1-r^{n+1}}{(1-r)^{2}}}-{\frac {(n+1)r^{n}}{1-r}}.}$$

For a geometric series containing only even powers of ${\displaystyle r}$ multiply by ${\displaystyle 1-r^{2}}$:

$${\displaystyle {\begin{aligned}(1-r^{2})\sum _{k=0}^{n}ar^{2k}&=a-ar^{2n+2}\\\sum _{k=0}^{n}ar^{2k}&={\frac {a(1-r^{2n+2})}{1-r^{2}}}\end{aligned}}}$$

Equivalently, take ${\displaystyle r^{2}}$ as the common ratio and use the standard formulation.

For a series with only odd powers of ${\displaystyle r}$,

$${\displaystyle {\begin{aligned}(1-r^{2})\sum _{k=0}^{n}ar^{2k+1}&=ar-ar^{2n+3}\\\sum _{k=0}^{n}ar^{2k+1}&={\frac {ar(1-r^{2n+2})}{1-r^{2}}}&={\frac {a(r-r^{2n+3})}{1-r^{2}}}\end{aligned}}}$$

An exact formula for the generalized sum ${\displaystyle G_{s}(n,r)}$ when ${\displaystyle s\in \mathbb {N} }$ is expanded by the Stirling numbers of the second kind as ^[6]

$${\displaystyle G_{s}(n,r)=\sum _{j=0}^{s}\left\lbrace {s \atop j}\right\rbrace x^{j}{\frac {d^{j}}{dx^{j}}}\left[{\frac {1-x^{n+1}}{1-x}}\right].}$$

Infinite series

An infinite geometric series is an infinite series whose successive terms have a common ratio. Such a series converges if and only if the absolute value of the common ratio is less than one (|r| < 1). Its value can then be computed from the finite sum formula

$${\displaystyle \sum _{k=0}^{\infty }ar^{k}=\lim _{n\to \infty }{\sum _{k=0}^{n}ar^{k}}=\lim _{n\to \infty }{\frac {a(1-r^{n+1})}{1-r}}={\frac {a}{1-r}}-\lim _{n\to \infty }{\frac {ar^{n+1}}{1-r}}}$$

Animation, showing convergence of partial sums of geometric progression ${\displaystyle \sum \limits _{k=0}^{n}q^{k}}$ (red line) to its sum ${\displaystyle {1 \over 1-q}}$ (blue line) for ${\displaystyle |q|<1}$.

Diagram showing the geometric series 1 + 1/2 + 1/4 + 1/8 + ⋯ which converges to 2.

Since:

$${\displaystyle r^{n+1}\to 0{\mbox{ as }}n\to \infty {\mbox{ when }}|r|<1.}$$

Then:

$${\displaystyle \sum _{k=0}^{\infty }ar^{k}={\frac {a}{1-r}}-0={\frac {a}{1-r}}}$$

For a series containing only even powers of ${\displaystyle r}$,

$${\displaystyle \sum _{k=0}^{\infty }ar^{2k}={\frac {a}{1-r^{2}}}}$$

and for odd powers only,

$${\displaystyle \sum _{k=0}^{\infty }ar^{2k+1}={\frac {ar}{1-r^{2}}}}$$

In cases where the sum does not start at k = 0,

$${\displaystyle \sum _{k=m}^{\infty }ar^{k}={\frac {ar^{m}}{1-r}}}$$

The formulae given above are valid only for |r| < 1. The latter formula is valid in every Banach algebra, as long as the norm of r is less than one, and also in the field of p-adic numbers if |r|_p < 1. As in the case for a finite sum, we can differentiate to calculate formulae for related sums. For example,

$${\displaystyle {\frac {d}{dr}}\sum _{k=0}^{\infty }r^{k}=\sum _{k=1}^{\infty }kr^{k-1}={\frac {1}{(1-r)^{2}}}}$$

This formula only works for |r| < 1 as well. From this, it follows that, for |r| < 1,

$${\displaystyle \sum _{k=0}^{\infty }kr^{k}={\frac {r}{\left(1-r\right)^{2}}}\,;\,\sum _{k=0}^{\infty }k^{2}r^{k}={\frac {r\left(1+r\right)}{\left(1-r\right)^{3}}}\,;\,\sum _{k=0}^{\infty }k^{3}r^{k}={\frac {r\left(1+4r+r^{2}\right)}{\left(1-r\right)^{4}}}}$$

Also, the infinite series 1/2 + 1/4 + 1/8 + 1/16 + ⋯ is an elementary example of a series that converges absolutely.

It is a geometric series whose first term is 1/2 and whose common ratio is 1/2, so its sum is

$${\displaystyle {\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots ={\frac {1/2}{1-(+1/2)}}=1.}$$

The inverse of the above series is 1/2 − 1/4 + 1/8 − 1/16 + ⋯ is a simple example of an alternating series that converges absolutely.

It is a geometric series whose first term is 1/2 and whose common ratio is −1/2, so its sum is

$${\displaystyle {\frac {1}{2}}-{\frac {1}{4}}+{\frac {1}{8}}-{\frac {1}{16}}+\cdots ={\frac {1/2}{1-(-1/2)}}={\frac {1}{3}}.}$$

Complex series

The summation formula for geometric series remains valid even when the common ratio is a complex number. In this case the condition that the absolute value of r be less than 1 becomes that the modulus of r be less than 1. It is possible to calculate the sums of some non-obvious geometric series. For example, consider the proposition

$${\displaystyle \sum _{k=0}^{\infty }{\frac {\sin(kx)}{r^{k}}}={\frac {r\sin(x)}{1+r^{2}-2r\cos(x)}}}$$

The proof of this comes from the fact that

$${\displaystyle \sin(kx)={\frac {e^{ikx}-e^{-ikx}}{2i}},}$$

which is a consequence of Euler's formula. Substituting this into the original series gives

$${\displaystyle \sum _{k=0}^{\infty }{\frac {\sin(kx)}{r^{k}}}={\frac {1}{2i}}\left[\sum _{k=0}^{\infty }\left({\frac {e^{ix}}{r}}\right)^{k}-\sum _{k=0}^{\infty }\left({\frac {e^{-ix}}{r}}\right)^{k}\right].}$$

This is the difference of two geometric series, and so it is a straightforward application of the formula for infinite geometric series that completes the proof.

Zeno of Elea (c.495 – c.430 BC)

2,500 years ago, Greek mathematicians had a problem when walking from one place to another: they thought^[7] that an infinitely long list of numbers greater than zero summed to infinity. Therefore, it was a paradox when Zeno of Elea pointed out that in order to walk from one place to another, you first have to walk half the distance, and then you have to walk half the remaining distance, and then you have to walk half of that remaining distance, and you continue halving the remaining distances an infinite number of times because no matter how small the remaining distance is you still have to walk the first half of it. Thus, Zeno of Elea transformed a short distance into an infinitely long list of halved remaining distances, all of which are greater than zero. And that was the problem: how can a distance be short when measured directly and also infinite when summed over its infinite list of halved remainders? The paradox revealed something was wrong with the assumption that an infinitely long list of numbers greater than zero summed to infinity.

Euclid of Alexandria (c.300 BC)

Elements of Geometry, Book IX, Proposition 35. "If there is any multitude whatsoever of continually proportional numbers, and equal to the first is subtracted from the second and the last, then as the excess of the second to the first, so the excess of the last will be to all those before it."

A geometric interpretation for the same case of common ratio r>1. (TOP) Represent the terms of a geometric series as the areas of overlapped similar triangles. (MIDDLE) From the largest to the smallest triangle, remove the overlapped left area portion (1/r) from the non-overlapped right area portion (1-1/r = (r-1)/r) and scale that non-overlapped trapezoid by r/(r-1) so its area is the same as the area of the original overlapped triangle. (BOTTOM) Calculate the area of the aggregate trapezoid as the area of the large triangle less the area of the empty small triangle at the large triangle's left tip. The large triangle is the largest overlapped triangle scaled by r/(r-1). The empty small triangle started as a but that area was transformed into a non-overlapped scaled trapezoid leaving an empty left area portion (1/r). However, that empty triangle of area a/r must also be scaled by r/(r-1) so its slope matches the slope of all the non-overlapped scaled trapezoids. Therefore, S_n = area of large triangle - area of empty small triangle = arⁿ⁺¹/(r-1) - a/(r-1) = a(rⁿ⁺¹-1)/(r-1).

Euclid's Elements of Geometry^[8] Book IX, Proposition 35, proof (of the proposition in adjacent diagram's caption):

Let AA', BC, DD', EF be any multitude whatsoever of continuously proportional numbers, beginning from the least AA'. And let BG and FH, each equal to AA', have been subtracted from BC and EF. I say that as GC is to AA', so EH is to AA', BC, DD'.

For let FK be made equal to BC, and FL to DD'. And since FK is equal to BC, of which FH is equal to BG, the remainder HK is thus equal to the remainder GC. And since as EF is to DD', so DD' to BC, and BC to AA' [Prop. 7.13], and DD' equal to FL, and BC to FK, and AA' to FH, thus as EF is to FL, so LF to FK, and FK to FH. By separation, as EL to LF, so LK to FK, and KH to FH [Props. 7.11, 7.13]. And thus as one of the leading is to one of the following, so (the sum of) all of the leading to (the sum of) all of the following [Prop. 7.12]. Thus, as KH is to FH, so EL, LK, KH to LF, FK, HF. And KH equal to CG, and FH to AA', and LF, FK, HF to DD', BC, AA'. Thus, as CG is to AA', so EH to DD', BC, AA'. Thus, as the excess of the second is to the first, so is the excess of the last is to all those before it. The very thing it was required to show.

The terseness of Euclid's propositions and proofs may have been a necessity. As is, the Elements of Geometry is over 500 pages of propositions and proofs. Making copies of this popular textbook was labor intensive given that the printing press was not invented until 1440. And the book's popularity lasted a long time: as stated in the cited introduction to an English translation, Elements of Geometry "has the distinction of being the world's oldest continuously used mathematical textbook." So being very terse was being very practical. The proof of Proposition 35 in Book IX could have been even more compact if Euclid could have somehow avoided explicitly equating lengths of specific line segments from different terms in the series. For example, the contemporary notation for geometric series (i.e., a + ar + ar² + ar³ + ... + arⁿ) does not label specific portions of terms that are equal to each other.

Also in the cited introduction the editor comments,

Most of the theorems appearing in the Elements were not discovered by Euclid himself, but were the work of earlier Greek mathematicians such as Pythagoras (and his school), Hippocrates of Chios, Theaetetus of Athens, and Eudoxus of Cnidos. However, Euclid is generally credited with arranging these theorems in a logical manner, so as to demonstrate (admittedly, not always with the rigour demanded by modern mathematics) that they necessarily follow from five simple axioms. Euclid is also credited with devising a number of particularly ingenious proofs of previously discovered theorems (e.g., Theorem 48 in Book 1).

To help translate the proposition and proof into a form that uses current notation, a couple modifications are in the diagram. First, the four horizontal line lengths representing the values of the first four terms of a geometric series are now labeled a, ar, ar², ar³ in the diagram's left margin. Second, new labels A' and D' are now on the first and third lines so that all the diagram's line segment names consistently specify the segment's starting point and ending point.

Here is a phrase by phrase interpretation of the proposition:

Similarly, here is a sentence by sentence interpretation of the proof:

Archimedes of Syracuse (c.287 – c.212 BC)

Archimedes' dissection of a parabolic segment into infinitely many triangles

Archimedes used the sum of a geometric series to compute the area enclosed by a parabola and a straight line. His method was to dissect the area into an infinite number of triangles.

Archimedes' Theorem states that the total area under the parabola is 4/3 of the area of the blue triangle.

Archimedes determined that each green triangle has 1/8 the area of the blue triangle, each yellow triangle has 1/8 the area of a green triangle, and so forth.

Assuming that the blue triangle has area 1, the total area is an infinite sum:

${\displaystyle 1\,+\,2\left({\frac {1}{8}}\right)\,+\,4\left({\frac {1}{8}}\right)^{2}\,+\,8\left({\frac {1}{8}}\right)^{3}\,+\,\cdots .}$

The first term represents the area of the blue triangle, the second term the areas of the two green triangles, the third term the areas of the four yellow triangles, and so on. Simplifying the fractions gives

${\displaystyle 1\,+\,{\frac {1}{4}}\,+\,{\frac {1}{16}}\,+\,{\frac {1}{64}}\,+\,\cdots .}$

This is a geometric series with common ratio 1/4 and the fractional part is equal to

${\displaystyle \sum _{n=0}^{\infty }4^{-n}=1+4^{-1}+4^{-2}+4^{-3}+\cdots ={4 \over 3}.}$

The sum is

${\displaystyle {\frac {1}{1-r}}\;=\;{\frac {1}{1-{\frac {1}{4}}}}\;=\;{\frac {4}{3}}.}$

This computation uses the method of exhaustion, an early version of integration. Using calculus, the same area could be found by a definite integral.

Nicole Oresme (c.1323 – 1382)

A two dimensional geometric series diagram Nicole Oresme used to determine that the infinite series 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 + ... converges to 2.

Among his insights into infinite series, in addition to his elegantly simple proof of the divergence of the harmonic series, Nicole Oresme^[9] proved that the series 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 + ... converges to 2. His diagram for his geometric proof, similar to the adjacent diagram, shows a two dimensional geometric series. The first dimension is horizontal, in the bottom row showing the geometric series S = 1/2 + 1/4 + 1/8 + 1/16 + ... , which is the geometric series with coefficient a = 1/2 and common ratio r = 1/2 that converges to S = a / (1-r) = (1/2) / (1-1/2) = 1. The second dimension is vertical, where the bottom row is a new coefficient a_T equal to S and each subsequent row above it is scaled by the same common ratio r = 1/2, making another geometric series T = 1 + 1/2 + 1/4 + 1/8 + ..., which is the geometric series with coefficient a_T = S = 1 and common ratio r = 1/2 that converges to T = a_T / (1-r) = S / (1-r) = a / (1-r) / (1-r) = (1/2) / (1-1/2) / (1-1/2) = 2.

Although difficult to visualize beyond three dimensions, Oresme's insight generalizes to any dimension d. Using the sum of the d−1 dimension of the geometric series as the coefficient a in the d dimension of the geometric series results in a d-dimensional geometric series converging to S^d / a = 1 / (1-r)^d within the range |r|<1. Pascal's triangle and long division reveals the coefficients of these multi-dimensional geometric series, where the closed form is valid only within the range |r|<1.

${\displaystyle {\begin{matrix}1\\1\quad 1\\1\quad 2\quad 1\\1\quad 3\quad 3\quad 1\\1\quad 4\quad 6\quad 4\quad 1\end{matrix}}}$

${\displaystyle d}$	${\displaystyle S^{d}}$ (closed form)	${\displaystyle S^{d}}$ (expanded form)
${\displaystyle 1}$	${\displaystyle 1/(1-r)}$	${\displaystyle 1+r+r^{2}+r^{3}+r^{4}+\cdots }$
${\displaystyle 2}$	${\displaystyle 1/(1-r)^{2}}$	${\displaystyle 1+2r+3r^{2}+4r^{3}+5r^{4}+\cdots }$
${\displaystyle 3}$	${\displaystyle 1/(1-r)^{3}}$	${\displaystyle 1+3r+6r^{2}+10r^{3}+15r^{4}+\cdots }$
${\displaystyle 4}$	${\displaystyle 1/(1-r)^{4}}$	${\displaystyle 1+4r+10r^{2}+20r^{3}+35r^{4}+\cdots }$

As an aside, instead of using long division, it is also possible to calculate the coefficients of the d-dimensional geometric series by integrating the coefficients of dimension d−1. This mapping from division by 1-r in the power series sum domain to integration in the power series coefficient domain is a discrete form of the mapping performed by the Laplace transform. MIT Professor Arthur Mattuck shows how to derive the Laplace transform from the power series in this lecture video,^[10] where the power series is a mapping between discrete coefficients and a sum and the Laplace transform is a mapping between continuous weights and an integral.

Economics

In economics, geometric series are used to represent the present value of an annuity (a sum of money to be paid in regular intervals).

For example, suppose that a payment of $100 will be made to the owner of the annuity once per year (at the end of the year) in perpetuity. Receiving $100 a year from now is worth less than an immediate $100, because one cannot invest the money until one receives it. In particular, the present value of $100 one year in the future is $100 / (1 + ${\displaystyle I}$ ), where ${\displaystyle I}$ is the yearly interest rate.

Similarly, a payment of $100 two years in the future has a present value of $100 / (1 + ${\displaystyle I}$)² (squared because two years' worth of interest is lost by not receiving the money right now). Therefore, the present value of receiving $100 per year in perpetuity is

${\displaystyle \sum _{n=1}^{\infty }{\frac {\$100}{(1+I)^{n}}},}$

which is the infinite series:

${\displaystyle {\frac {\$100}{(1+I)}}\,+\,{\frac {\$100}{(1+I)^{2}}}\,+\,{\frac {\$100}{(1+I)^{3}}}\,+\,{\frac {\$100}{(1+I)^{4}}}\,+\,\cdots .}$

This is a geometric series with common ratio 1 / (1 + ${\displaystyle I}$ ). The sum is the first term divided by (one minus the common ratio):

${\displaystyle {\frac {\$100/(1+I)}{1-1/(1+I)}}\;=\;{\frac {\$100}{I}}.}$

For example, if the yearly interest rate is 10% (${\displaystyle I}$ = 0.10), then the entire annuity has a present value of $100 / 0.10 = $1000.

This sort of calculation is used to compute the APR of a loan (such as a mortgage loan). It can also be used to estimate the present value of expected stock dividends, or the terminal value of a financial asset assuming a stable growth rate.

Fractal geometry

The interior of the Koch snowflake is a union of infinitely many triangles. In the study of fractals, geometric series often arise as the perimeter, area, or volume of a self-similar figure.

If |r| < 1, representing each geometric series term as the area of a similar triangle creates a fractal in which more zooming in always reveals more similar triangles.

The area inside the Koch snowflake can be described as the union of infinitely many equilateral triangles (see figure). Each side of the green triangle is exactly 1/3 the size of a side of the large blue triangle, and therefore has exactly 1/9 the area. Similarly, each yellow triangle has 1/9 the area of a green triangle, and so forth. Taking the blue triangle as a unit of area, the total area of the snowflake is

${\displaystyle 1\,+\,3\left({\frac {1}{9}}\right)\,+\,12\left({\frac {1}{9}}\right)^{2}\,+\,48\left({\frac {1}{9}}\right)^{3}\,+\,\cdots .}$

The first term of this series represents the area of the blue triangle, the second term the total area of the three green triangles, the third term the total area of the twelve yellow triangles, and so forth. Excluding the initial 1, this series is geometric with constant ratio r = 4/9. The first term of the geometric series is a = 3(1/9) = 1/3, so the sum is

${\displaystyle 1\,+\,{\frac {a}{1-r}}\;=\;1\,+\,{\frac {\frac {1}{3}}{1-{\frac {4}{9}}}}\;=\;{\frac {8}{5}}.}$

Thus the Koch snowflake has 8/5 of the area of the base triangle.

Integration

The derivative of ${\displaystyle f(x)=\arctan(u(x)){\text{ is }}f'(x)=u'(x)/(1+[u(x)]^{2})}$ because,^[11] letting ${\displaystyle y{\text{ and }}u{\text{ represent }}f(x){\text{ and }}u(x),}$

${\displaystyle {\begin{aligned}y&=\arctan(u)&&\quad {\text{ implies }}\\u&=\tan(y)&&\quad {\text{ in the range }}-\pi /2<y<\pi /2{\text{ and }}\\u'&=\sec ^{2}y\cdot y'&&\quad {\text{ by applying the quotient rule to }}\tan(y)=\sin(y)/\cos(y),\\y'&=u'/\sec ^{2}y&&\quad {\text{ by dividing both sides by }}\sec ^{2}y,\\&=u'/(1+\tan ^{2}y)&&\quad {\text{ by using the trigonometric identity derived by dividing }}\sin ^{2}y+\cos ^{2}y=1{\text{ by }}\cos ^{2}y,\\&=u'/(1+u^{2})&&\quad {\text{ by recalling }}u=\tan(y).\end{aligned}}}$

Therefore, letting ${\displaystyle u(x)=x,\arctan(x)}$ is the integral

${\displaystyle {\begin{aligned}\arctan(x)&=\int {\frac {dx}{1+x^{2}}}\quad &&{\text{in the range }}-\pi /2<\arctan(x)<\pi /2,\\&=\int {\frac {dx}{1-(-x^{2})}}\quad &&{\text{by writing integrand as closed form of geometric series with }}r=-x^{2},\\&=\int \left(1+\left(-x^{2}\right)+\left(-x^{2}\right)^{2}+\left(-x^{2}\right)^{3}+\cdots \right)dx\quad &&{\text{by writing geometric series in expanded form}},\\&=\int \left(1-x^{2}+x^{4}-x^{6}+\cdots \right)dx\quad &&{\text{by calculating the sign and power of each term in integrand}},\\&=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-{\frac {x^{7}}{7}}+\cdots \quad &&{\text{by integrating each term}},\\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}x^{2n+1}\quad &&{\text{by writing series in generator form}},\end{aligned}}}$

which is called Gregory's series and is commonly attributed to Madhava of Sangamagrama (c. 1340 – c. 1425).

Geometric series

A compact illustration of different perspectives on the geometric series to make perspective switching easier. Quadrant A1 contains the why perspective briefly describing why geometric series matter. Quadrant A2 contains the how perspective listing a snippet of Julia code that computes the geometric series. Quadrant A3 contains the what (algebra) perspective with the algebraic proof of the geometric series closed form formula. Quadrant A4 contains the what (geometry) perspective showing three steps of the geometric proof of the geometric series closed form formula. And the right margin shows the timeline of some of the historical insights about the geometric series.

The geometric series has two degrees of freedom: one for its coefficient a and another for its common ratio r. In the map of polynomials, the big red circle represents the set of all geometric series.

Power series

Like the geometric series, the power series has one degree of freedom for its common ratio r (along the x-axis) but has n+1 degrees of freedom for its coefficients (along the y-axis), where n represents the power of the last term in the partial series. In the map of polynomials, the big blue circle represents the set of all power series.

Taylor series

This section is an excerpt from Taylor series.[edit]

As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows sin x and its Taylor approximations by polynomials of degree 1, 3, 5, 7, 9, 11, and 13 at x = 0.

In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century.

The partial sum formed by the first n + 1 terms of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally more accurate as n increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the infinite sequence of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent. A function is analytic at a point x if it is equal to the sum of its Taylor series in some open interval (or open disk in the complex plane) containing x. This implies that the function is analytic at every point of the interval (or disk).

Binary encoded numbers

Main article: Binary number

Main article: Single-precision floating-point format

Zeno of Elea's geometric series with coefficient a=1/2 and common ratio r=1/2 is the foundation of binary encoded approximations of fractions in digital computers. Concretely, the geometric series written in its normalized vector form is s/a = [1 1 1 1 1 …][1 r r² r³ r⁴ …]^T. Keeping the column vector of basis functions [1 r r² r³ r⁴ …]^T the same but generalizing the row vector [1 1 1 1 1 …] so that each entry can be either a 0 or a 1 allows for an approximate encoding of any fraction. For example, the value v = 0.34375 is encoded as  v/a = [0 1 0 1 1 0 …][1 r r² r³ r⁴ …]^T where coefficient a = 1/2 and common ratio r = 1/2. Typically, the row vector is written in the more compact binary form v = 0.010110 which is 0.34375 in decimal.

Similarly, the geometric series with coefficient a=1 and common ratio r=2 is the foundation for binary encoded integers in digital computers. Again, the geometric series written in its normalized vector form is s/a = [1 1 1 1 1 …][1 r r² r³ r⁴ …]^T. Keeping the column vector of basis functions [1 r r² r³ r⁴ …]^T the same but generalizing the row vector [1 1 1 1 1 …] so that each entry can be either a 0 or a 1 allows for an encoding of any integer. For example, the value v = 151 is encoded as  v/a = [1 1 1 0 1 0 0 1 0 …][1 r r² r³ r⁴ r⁵ r⁶ r⁷ r⁸ …]^T where coefficient a = 1 and common ratio r = 2. Typically, the row vector is written in reverse order (so that the most significant bit is first) in the more compact binary form v = …010010111 = 10010111 which is 151 in decimal.

Bit fields for encoding a 32-bit floating point number according to IEEE 754 standard.

As shown in the adjacent figure, the standard binary encoding of a 32-bit floating point number is a combination of a binary encoded integer and a binary encoded fraction, beginning at the most significant bit with

• the sign bit, followed by
• an 8-bit integer exponent field with an assumed offset of 127 (so a value of 127 represents an exponent value of 0) and with a base of 2 meaning that the exponent value specifies a bit shift of the fraction field, followed by
• a 23-bit fraction field with an assumed but not encoded 1 serving as the fraction's most significant nonzero bit which would be in bit position 23 if it were encoded.

Building upon the previous example of 0.34375 having binary encoding of 0.010110, a floating point encoding (according to the IEEE 754 standard) of 0.34375 is

• the sign bit which is 0 because the number is not negative,
• an 8-bit integer exponent field which must specify a shift that counters the 2 bit left shift to get the original binary encoding from 0.010110 to 1.0110, and that counter shift to recover the original binary encoding is a right shift of 2 bits which is specified by an exponent value of 125 (because 125 − 127 = -2 which is a right shift of 2 bits) which in binary is 0111 1101,
• a 23-bit fraction field: .0110 0000 0000 0000 0000 000.

Although encoding floating point numbers by hand like this is possible, letting a computer do it is easier and less error prone. The following Julia code confirms the hand calculated floating point encoding of the number 0.34375:

julia> bitstring(Float32(0.34375))
"00111110101100000000000000000000"

Laurent series

This section is an excerpt from Laurent series.[edit]

A Laurent series is defined with respect to a particular point ${\displaystyle c}$ and a path of integration γ. The path of integration must lie in an annulus, indicated here by the red color, inside which ${\displaystyle f(z)}$ is holomorphic (analytic).

In mathematics, the Laurent series of a complex function ${\displaystyle f(z)}$ is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Karl Weierstrass may have discovered it first in a paper written in 1841, but it was not published until after his death.^[12]

Matrix polynomial

Matrix exponential

This section is an excerpt from Matrix exponential.[edit]

In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group.

Let X be an n×n real or complex matrix. The exponential of X, denoted by e^X or exp(X), is the n×n matrix given by the power series

$${\displaystyle e^{X}=\sum _{k=0}^{\infty }{\frac {1}{k!}}X^{k}}$$

where ${\displaystyle X^{0}}$ is defined to be the identity matrix ${\displaystyle I}$ with the same dimensions as ${\displaystyle X}$.^[15] The series always converges, so the exponential of X is well-defined.

Equivalently,

$${\displaystyle e^{X}=\lim _{k\rightarrow \infty }\left(I+{\frac {X}{k}}\right)^{k}}$$

where I is the n×n identity matrix.

When X is an n×n diagonal matrix then exp(X) will be an n×n diagonal matrix with each diagonal element equal to the ordinary exponential applied to the corresponding diagonal element of X.

Definitions of theta functions

In the Geometric series the summands take the exponential course in relation to the sum index. The exponent in the summands, which is dependent on the sum index, has a linear course with respect to the index. However, if the exponent in the elements of the sequence instead assumes a square course with respect to the sum index, then the sequence members themselves follow course of a Gaussian bell curve, and then the values are of the affected infinite series can not be displayed in an elementary way. Sums with a square course of the exponent in the sequence members with respect to the sum index assume elliptic theta function values for the representation. And these values can be represented with both the Jacobian theta functions and with the Neville theta functions as well.

The theta functions mentioned are according to Edmund Taylor Whittaker and George Neville WatsonTheta functions^[16][17][18] defined as follows:

${\displaystyle \vartheta _{00}(v;w)=\prod _{n=1}^{\infty }(1-w^{2n})[1+2\cos(2v)w^{2n-1}+w^{4n-2}]}$

${\displaystyle \theta _{d}(y;k)=\prod _{n=1}^{\infty }{\bigl [}1+q(k)^{2n-1}{\bigr ]}^{-2}{\bigl \{}1+2q(k)^{2n-1}\cos {\bigl [}\pi \,y\div K(k){\bigr ]}+q(k)^{4n-2}{\bigr \}}}$

The function ${\displaystyle q(k)}$ is called the Elliptic Nome and the function ${\displaystyle K(k)}$ stands for the complete Elliptic Integral of first kind:

${\displaystyle q(k)=\exp {\bigl [}-\pi \,K({\sqrt {1-k^{2}}}\,)\div K(k){\bigr ]}=\exp {\bigl [}-\pi \,K'(k)\div K(k){\bigr ]}}$

${\displaystyle K(\varepsilon )=\int _{0}^{1}{\bigl [}(1-z^{2})(1-\varepsilon ^{2}z^{2}){\bigr ]}^{-1/2}\,\mathrm {d} z}$

${\displaystyle K'(\varepsilon )=K({\sqrt {1-\varepsilon ^{2}}}\,)}$

The Greek letter designation ${\displaystyle \vartheta _{00}}$ represents the so called Theta Nullwert Function whereas the designation ${\displaystyle \psi _{H}}$ stands for the Hermite elliptic Psi Function:

${\displaystyle \vartheta _{00}(w)=\prod _{n=1}^{\infty }(1-w^{2n})(1+w^{2n-1})^{2}}$

${\displaystyle \psi _{H}(w)=\prod _{n=1}^{\infty }{\frac {1-w^{2n-1}}{1+w^{2n-1}}}}$

Identities for the sum series

This interconnection is given:

${\displaystyle q(\varepsilon )=\exp {\bigl [}-\pi \,K'(\varepsilon )\div K(\varepsilon ){\bigr ]}}$	${\displaystyle q'(\varepsilon )=\exp {\bigl [}-\pi \,K(\varepsilon )\div K'(\varepsilon ){\bigr ]}}$	${\displaystyle Q'=\exp {\bigl [}\pi ^{2}\div \ln(Q){\bigr ]}}$
Continuation of the identity by using the Neville Theta Function:
${\displaystyle \sum _{n=-\infty }^{\infty }Q^{(n+R)^{2}}={\frac {\vartheta _{00}(Q)}{\vartheta _{00}(Q')}}\,\vartheta _{00}{\bigl (}\pi \,R\,;Q'{\bigr )}=\vartheta _{00}(Q)\,\theta _{d}{\bigl \{}2\,R\,K{\bigl [}\psi _{H}(Q)^{4}{\bigr ]};\psi _{H}(Q)^{4}{\bigr \}}}$
Continuation of the identity by using the Poisson Summation Formula:
${\displaystyle \sum _{n=-\infty }^{\infty }Q^{(n+R)^{2}}={\frac {\vartheta _{00}(Q)}{\vartheta _{00}(Q')}}\,\vartheta _{00}{\bigl (}\pi \,R\,;Q'{\bigr )}={\biggl [}{\frac {\pi }{\ln(1\div Q)}}{\biggr ]}^{1/2}\,\vartheta _{00}{\bigl (}\pi \,R\,;Q'{\bigr )}}$

The Neville theta functions themselves were researched by the mathematician Eric Harold Neville from England. In general, by hiking integer numbered steps of ${\displaystyle R}$ the result must always be the same value over and over again. This very phenomenon is called periodicity of functions. And this criterion is fulfilled by the theta formulas mentioned above. If R is set to zero, then the primary standardized theta function value ${\displaystyle \vartheta _{00}(Q)}$ must be the result, the base value ${\displaystyle Q}$ must be used as the Nome entry of the Theta Nullwert function. And if the value ${\displaystyle R}$ does not result in a whole number but rather a fractional number, then the resulting value of the sum series must be related to the value of the sum series with ${\displaystyle R}$ equal to zero by the factor which is given by the associated incomplete theta function of the Complementary Nome, i.e. by the value ${\displaystyle \vartheta _{00}(\pi \,R;Q')\div \vartheta _{00}(Q')}$ exactly. The formula in the bottom line of the formula table shown follows directly from the laws described in the last two sentences mentioned. And the formula with Neville's theta function results directly from this. The Pythagorean complementary module must be entered as the module. And this is created accurately by the fourth power of the Hermite Elliptic Psi Function ${\displaystyle \psi _{H}(Q)^{4}}$. By using the Poisson's empirical formula, simplifications can be made in the formula in the bottom line of the table.

Calculation examples with the Jacobi theta

This formula about the Jacobi theta function is valid:

${\displaystyle \sum _{n=-\infty }^{\infty }x^{an^{2}+bn+c}=\exp {\biggl [}{\frac {1}{4a}}(4ac-b^{2})\ln(x){\biggr ]}{\biggl [}{\frac {\pi }{a\ln(1\div x)}}{\biggr ]}^{1/2}\vartheta _{00}{\biggl \{}{\frac {\pi b}{2a}};\exp {\biggl [}{\frac {-\pi ^{2}}{a\ln(1\div x)}}{\biggr ]}{\biggr \}}}$

Hierfür soll gelten: ${\displaystyle (0<x<1)\,\,\cap \,\,(a\in \mathbb {R} ^{+})}$

For the now mentioned sum series following two calculation examples shall be executed in detail:

${\displaystyle \sum _{n=-\infty }^{\infty }{\bigl (}{\frac {7}{11}}{\bigr )}^{5n^{2}+3n+2}={\bigl (}{\frac {7}{11}}{\bigr )}^{31/20}{\bigl (}{\frac {\pi }{5}}{\bigr )}^{1/2}\ln {\bigl (}{\frac {11}{7}}{\bigr )}^{-1/2}\vartheta _{00}{\biggl \{}{\frac {3\pi }{10}};\exp {\biggl [}-{\frac {\pi ^{2}}{5}}\ln {\bigl (}{\frac {11}{7}}{\bigr )}^{-1}{\biggr ]}{\biggr \}}}$

${\displaystyle \sum _{n=-\infty }^{\infty }{\bigl (}{\frac {23}{29}}{\bigr )}^{19n^{2}+17n+13}={\bigl (}{\frac {23}{29}}{\bigr )}^{699/76}{\bigl (}{\frac {\pi }{19}}{\bigr )}^{1/2}\ln {\bigl (}{\frac {29}{23}}{\bigr )}^{-1/2}\vartheta _{00}{\biggl \{}{\frac {17\pi }{38}};\exp {\biggl [}-{\frac {\pi ^{2}}{19}}\ln {\bigl (}{\frac {29}{23}}{\bigr )}^{-1}{\biggr ]}{\biggr \}}}$