geometric series

In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series

:$\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots$

is geometric, because each successive term can be obtained by multiplying the previous term by $1/2$. In general, a geometric series is written as $a + ar + ar^2 + ar^3 + ...$, where $a$ is the coefficient of each term and $r$ is the common ratio between adjacent terms. The geometric series had an important role in the early development of calculus, is used throughout mathematics, and can serve as an introduction to frequently used mathematical tools such as the Taylor series, the complex Fourier series, and the matrix exponential.

The name geometric series indicates each term is the geometric mean of its two neighboring terms, similar to how the name arithmetic series indicates each term is the arithmetic mean of its two neighboring terms. The sequence of geometric series terms (without any of the additions) is called a '' geometric sequence'' or ''geometric progression''.

Formulation

Coefficient ''a''

The geometric series ''a'' + ''ar'' + ''ar''² + ''ar''³ + ... is written in expanded form.Riddle, Douglas F. ''Calculus and Analytic Geometry, Second Edition'' Belmont, California, Wadsworth Publishing, p. 566, 1970. Every coefficient in the geometric series is the same. In contrast, the power series written as ''a''₀ + ''a''₁''r'' + ''a''₂''r''² + ''a''₃''r''³ + ... 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 of the geometric series written as

:$\sum^_$ ''ar''^k

and a closed form of the geometric series written as

:$\frac \text , r, <1.$

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'' + ''ar''² + ''ar''³ + ... , 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'' + ''ar''² + ''ar''³ + ''ar''⁴ + ... in its normalized form
:''s'' / ''a'' = 1 + ''r'' + ''r''² + ''r''³ + ''r''⁴ + ... or in its normalized vector form
:''s'' / ''a'' = 1 1 1 1 ...1 ''r'' ''r''² ''r''³ ''r''⁴ ...]^T or in its normalized partial series form
:''s''_n / ''a'' = 1 + ''r'' + ''r''² + ''r''³ + ''r''⁴ + ... + ''r''ⁿ, 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 geometric series ''a'' + ''ar'' + ''ar''² + ''ar''³ + ... 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'' + ''r''² + ''r''³ + ... 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'' + ''ar''² + ''ar''³ + ... has the geometric progression (also called the geometric sequence) ''a'', ''ar'', ''ar''², ''ar''³, ... 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^i''θ'' 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^i''θ'', the expanded form of the geometric series is ''a'' + ''a'', ''r'', e^i''θ'' + ''a'', ''r'', ²e^i2''θ'' + ''a'', ''r'', ³e^i3''θ'' + ... Modeling the angle ''θ'' as linearly increasing over time at the rate of some angular frequency ''ω''₀ (in other words, making the substitution ''θ'' = ''ω''₀''t''), the expanded form of the geometric series becomes ''a'' + ''a'', ''r'', e^{i''ω''₀''t''} + ''a'', ''r'', ²e^{i2''ω''₀''t''} + ''a'', ''r'', ³e^{i3''ω''₀''t''} + ... , 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 ''ω''₀. 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.

Sum

The sum of the first ''n'' terms of a geometric series, up to and including the ''r'' ^n-1 term, is given by the closed-form formula:

$\begin s_n &= ar^0 + ar^1 + \cdots + ar^\\ &= \sum_^ ar^k = \sum_^ ar^\\ &= \begin a\left(\frac\right), \text r \neq 1\\ an, \text r = 1 \end \end$

where is the common ratio. One can derive that closed-form formula for the partial sum, ''s''_n, by subtracting out the many self-similar terms as follows:
$\begin s_n &= ar^0 + ar^1 + \cdots + ar^,\\ rs_n &= ar^1 + ar^2 + \cdots + ar^,\\ s_n - rs_n &= ar^0 - ar^,\\ s_n\left(1-r\right) &= a\left(1-r^\right),\\ s_n &= a\left(\frac\right), \text r \neq 1. \end$

As approaches infinity, the absolute value of must be less than one for the series to converge. The sum then becomes

$\begin s &= a+ar+ar^2+ar^3+ar^4+\cdots\\ &= \sum_^\infty ar^ = \sum_^\infty ar^\\ &= \frac, \text , r, <1. \end$

The formula also holds for complex , with the corresponding restriction that the modulus of is strictly less than one.

As an aside, the question of whether an infinite series converges is fundamentally a question about the distance between two values: given enough terms, does the value of the partial sum get arbitrarily close to the finite value it is approaching? In the above derivation of the closed form of the geometric series, the interpretation of the distance between two values is the distance between their locations on the number line. That is the most common interpretation of the distance between two values. However, the p-adic metric, which has become a critical notion in modern number theory, offers a definition of distance such that the geometric series 1 + 2 + 4 + 8 + ... with ''a'' = 1 and ''r'' = 2 actually does converge to ''a'' / (1 - ''r'') = 1 / (1 - 2) = -1 even though ''r'' is outside the typical convergence range , ''r'', < 1.

Proof of convergence

We can prove that the geometric series converges using the sum formula for a geometric progression:
:$\begin 1 + r + r^2 + r^3 + \cdots \ &= \lim_ \left(1 + r + r^2 + \cdots + r^n\right) \\ &= \lim_ \frac. \end$
The second equality is true because if $, r, < 1,$ then $r^ \to 0$ as $n \to \infty$ and
:$\begin (1 + r + r^2 + \cdots + r^n)(1 - r) &= ((1-r) + (r - r^2) + (r^2 - r^3) + ... + (r^n - r^))\\ &= (1 + (-r + r) + ( -r^2 + r^2) + ... + (-r^n + r^n) - r^)\\ &= 1-r^. \end$

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 = ''ar''ⁿ⁺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

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 ''ar''ⁿ and the previous series remainder is ''s'' - ''s''_n-1 = ''ar''ⁿ / (1 - ''r'')), this measure of the convergence rate of the geometric series is ''ar''ⁿ / (''ar''ⁿ / (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 - ''r''² 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 - ''r''²) = (1 - , ''r'', ), or scale = (1 - , ''r'', ) / (1 - ''r''²) = (1 + ''r'') / (1 - ''r''²) = (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 ''r''² 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 - ''r''², 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:

$\sum_^ ar^ = ar^0+ar^1+ar^2+ar^3+\cdots+ar^.$

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

$\begin (1-r) \sum_^ ar^ & = (1-r)(ar^0 + ar^1+ar^2+ar^3+\cdots+ar^) \\ & = ar^0 + ar^1+ar^2+ar^3+\cdots+ar^ - ar^1-ar^2-ar^3-\cdots-ar^ - ar^n \\ & = a - ar^n \end$

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:

$\sum_^ ar^ = \frac.$

; Related formulas

If one were to begin the sum not from k=1 or 0 but from a different value, say , then

$\begin \sum_^n ar^ &= \begin \frac & \textr \neq 1 \\ a(n-m+1) & \textr = 1 \end\\ \sum_^n ar^k &= \begin a(n-m+1) & \textr = 1 \\ \frac & \textr \neq 1 \end\end$

Differentiating this formula with respect to allows us to arrive at formulae for sums of the form

$G_s(n, r) := \sum_^n k^s r^k.$

For example:

$\frac\sum_^nr^k = \sum_^n kr^= \frac-\frac.$

For a geometric series containing only even powers of multiply by :

$\begin (1-r^2) \sum_^ ar^ &= a-ar^\\ \sum_^ ar^ &= \frac \end$

Equivalently, take as the common ratio and use the standard formulation.

For a series with only odd powers of ,
$\begin (1-r^2) \sum_^ ar^ &= ar-ar^\\ \sum_^ ar^ &= \frac &= \frac \end$

An exact formula for the generalized sum $G_s(n, r)$ when $s \in \mathbb$ is expanded by the Stirling numbers of the second kind as

G_s(n, r) = \sum_^s \left\lbrace\right\rbrace x^j \frac\left frac\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 ( < 1). Its value can then be computed from the finite sum formula

:$\sum_^\infty ar^k = \lim_ = \lim_\frac= \frac - \lim_$

Since:
:$r^ \to 0 \mbox n \to \infty \mbox , r, < 1.$

Then:
:$\sum_^\infty ar^k = \frac - 0 = \frac$

For a series containing only even powers of $r$,

:$\sum_^\infty ar^ = \frac$

and for odd powers only,

:$\sum_^\infty ar^ = \frac$

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

:$\sum_^\infty ar^k=\frac$
The formulae given above are valid only for  < 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 _''p'' < 1. As in the case for a finite sum, we can differentiate to calculate formulae for related sums.
For example,

:$\frac\sum_^\infty r^k = \sum_^\infty kr^= \frac$

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

:$\sum_^ k r^k = \frac \,;\, \sum_^ k^2 r^k = \frac \, ; \, \sum_^ k^3 r^k = \frac$

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
:$\frac12+\frac14+\frac18+\frac+\cdots=\frac = 1.$

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
:$\frac12-\frac14+\frac18-\frac+\cdots=\frac = \frac13.$

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

: $\sum_^ \frac = \frac$

The proof of this comes from the fact that
: $\sin(kx) = \frac ,$
which is a consequence of Euler's formula. Substituting this into the original series gives
: \sum_^ \frac = \frac \left \sum_^ \left( \frac \right)^k - \sum_^ \left(\frac\right)^k\right/math>.

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.

History

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 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)

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

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'' + ''ar''² + ''ar''³ + ... + ''ar''ⁿ) 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 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:
:$1 \,+\, 2\left(\frac\right) \,+\, 4\left(\frac\right)^2 \,+\, 8\left(\frac\right)^3 \,+\, \cdots.$

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
:$1 \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots.$

This is a geometric series with common ratio and the fractional part is equal to
:$\sum_^\infty 4^ = 1 + 4^ + 4^ + 4^ + \cdots = .$

The sum is
:$\frac\;=\;\frac\;=\;\frac.$

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)

Among his insights into infinite series, in addition to his elegantly simple proof of the divergence of the harmonic series, Nicole Oresme 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.