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Descriptions

Definition, common ratio, and examples

The geometric series is an infinite series derived from a special type of sequence called a geometric progression. This series has two parameters known as the initial term $a$ and common ratio $r$. The geometric series can be described as: $$a+ar+ar^{2}+ar^{3}+\dots =\sum _{k=0}^{\infty }ar^{k}.$$ Truncating the geometric series into several terms is called finite geometric series, that is: $$a+ar+ar^{2}+ar^{3}+\dots +ar^{n}=\sum _{k=0}^{n}ar^{k}.$$

The common ratio is the ratio of any term with the previous term in the series, or it can be interpreted as a multiplier used to calculate each next term in the series from the previous term. That is, $$r={\frac {ar^{k+1}}{ar^{k}}},$$

When $r>1$ it is often called a growth rate or rate of expansion. When $0<r<1$ it is often called a decay rate or shrink rate, where the idea that it is a "rate" comes from interpreting $k$ as a sort of discrete time variable. When an application area has specialized vocabulary for specific types of growth, expansion, shrinkage, and decay, that vocabulary will also often be used to name $r$ parameters of geometric series. In economics, for instance, rates of increase and decrease of price levels are called inflation rates and deflation rates, while rates of increase in values of investments include rates of return and interest rates.

Convergence of the series and its proof

The convergence of the infinite sequence of partial sums of the infinite geometric series depends on the magnitude of the common ratio $r$ alone:

• If $\vert r\vert <1$, the terms of the series approach zero (becoming smaller and smaller in magnitude) and the sequence of partial sums $S_{n}$ converge to a limit value of ${\frac {a}{1-r}};$ proof is provided below.
• If $\vert r\vert >1$, the terms of the series become larger and larger in magnitude and the partial sums of the terms also get larger and larger in magnitude, so the series diverges.
• If $\vert r\vert =1$, the terms of the series become no larger or smaller in magnitude and the sequence of partial sums of the series does not converge. When $r=1$, all the terms of the series are the same and the $|S_{n}|$ grow to infinity. When $r=-1$, the terms take two values $a$ and $-a$ alternately, and therefore the sequence of partial sums of the terms oscillates between the two values $a$ and 0. Consider, for example, Grandi's series: $1-1+1-1+1+...$. Partial sums of the terms oscillate between 1 and 0; the sequence of partial sums does not converge. When $r=i$ and $a=1$, the partial sums circulate periodically among the values $1,1+i,i,0,1,1+i,i,0,\ldots$, never converging to a limit. Generally when $r=e^{2\pi i/\tau }$ for any integer $\tau$ and with any $a\neq 0$, the partial sums of the series will circulate indefinitely with a period of $\tau$, never converging to a limit.

The rate of convergence shows how the sequence quickly approaches its limit. In the case of the geometric series—the relevant sequence is $S_{n}$ and its limit is $S$—the rate and order are found via $$\lim _{n\rightarrow \infty }{\frac {\left|S_{n+1}-S\right|}{\left|S_{n}-S\right|^{q}}},$$ where $q$ represents the order of convergence. Using ${\textstyle |S_{n}-S|=\left|{\frac {ar^{n+1}}{1-r}}\right|}$ and choosing the order of convergence $q=1$ gives: $$\lim _{n\rightarrow \infty }{\frac {\left|{\frac {ar^{n+2}}{1-r}}\right|}{\left|{\frac {ar^{n+1}}{1-r}}\right|^{1}}}=|r|.$$ When the series converges, the rate of convergence gets slower as $|r|$ approaches $1$. The pattern of convergence also depends on the sign or complex argument of the common ratio. If $r>0$ and $|r|<1$ then terms all share the same sign and the partial sums of the terms approach their eventual limit monotonically. If $r<0$ and $|r|<1$, adjacent terms in the geometric series alternate between positive and negative, and the partial sums $S_{n}$ of the terms oscillate above and below their eventual limit $S$. For complex $r$ and $|r|<1,$ the $S_{n}$ converge in a spiraling pattern.

The convergence is proved as follows. The partial sum of the first $n+1$ terms of a geometric series, up to and including the $r^{n}$ term, $$S_{n}=ar^{0}+ar^{1}+\cdots +ar^{n}=\sum _{k=0}^{n}ar^{k},$$ is given by the closed form $$S_{n}={\begin{cases}a(n+1)&r=1\\a\left({\frac {1-r^{n+1}}{1-r}}\right)&{\text{otherwise}}\end{cases}}$$ where $r$ is the common ratio. The case $r=1$ is just simple addition, a case of an arithmetic series. The formula for the partial sums $S_{n}$ with $r\neq 1$ can be derived as follows: $${\begin{aligned}S_{n}&=ar^{0}+ar^{1}+\cdots +ar^{n},\\rS_{n}&=ar^{1}+ar^{2}+\cdots +ar^{n+1},\\S_{n}-rS_{n}&=ar^{0}-ar^{n+1},\\S_{n}\left(1-r\right)&=a\left(1-r^{n+1}\right),\\S_{n}&=a\left({\frac {1-r^{n+1}}{1-r}}\right),\end{aligned}}$$ for $r\neq 1$. As $r$ approaches 1, polynomial division or L'Hospital's rule recovers the case $S_{n}=a(n+1)$.

Proof without words of the formula for the sum of a geometric series – if |r| < 1 and n → ∞, the rⁿ term vanishes, leaving S_∞ = ⁠a/1 − r⁠. This figure uses a slightly different convention for S_n than the main text, shifted by one term.

As $n$ approaches infinity, the absolute value of r must be less than one for this sequence of partial sums to converge to a limit. When it does, the series converges absolutely. The infinite series then becomes$${\begin{aligned}S&=a+ar+ar^{2}+ar^{3}+ar^{4}+\cdots \\&=\lim _{n\rightarrow \infty }S_{n}\\&=\lim _{n\rightarrow \infty }{\frac {a(1-r^{n+1})}{1-r}}\\&={\frac {a}{1-r}}-{\frac {a}{1-r}}\lim _{n\rightarrow \infty }r^{n+1}\\&={\frac {a}{1-r}},\end{aligned}}$$ for $|r|<1$.

This convergence result is widely applied to prove the convergence of other series as well, whenever those series's terms can be bounded from above by a suitable geometric series; that proof strategy is the basis for the ratio test and root test for the convergence of infinite series.

Connection to the power series

Like the geometric series, a power series $a_{0}+a_{1}x+a_{2}x^{2}+\ldots$ has one parameter for a common variable raised to successive powers, denoted $x$ here, corresponding to the geometric series's r, but it has additional parameters $a_{0},a_{1},a_{2},\ldots ,$ one for each term in the series, for the distinct coefficients of each $x^{0},x^{1},x^{2},\ldots$, rather than just a single additional parameter $a$ for all terms, the common coefficient of $r^{k}$ in each term of a geometric series.

The geometric series can therefore be considered a class of power series in which the sequence of coefficients satisfies $a_{k}=a$ for all $k$ and $x=r$. This special class of power series plays an important role in mathematics, for instance for the study of ordinary generating functions in combinatorics and the summation of divergent series in analysis. Many other power series can be written as transformations and combinations of geometric series, making the geometric series formula a convenient tool for calculating formulas for those power series as well.

As a power series, the geometric series has a radius of convergence of 1. This could be seen as a consequence of the Cauchy–Hadamard theorem and the fact that ${\textstyle \lim _{n\rightarrow \infty }{\sqrt[{n}]{a}}=1}$ for any $a$ or as a consequence of the ratio test for the convergence of infinite series, with ${\textstyle \lim _{n\rightarrow \infty }|ar^{n+1}|/|ar^{n}|=|r|}$ implying convergence only for $|r|<1.$ However, both the ratio test and the Cauchy–Hadamard theorem are proven using the geometric series formula as a logically prior result, so such reasoning would be subtly circular.

Applications

In economics, specifically in mathematical finance, geometric series are used in time value of money; that is to represent the present values of perpetual annuities, sums of money to be paid each year indefinitely into the future. This sort of calculation is used to compute the annual percentage rate 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. However, the assumption that interest rates are constant is generally incorrect and payments are unlikely to continue forever since the issuer of the perpetual annuity may lose its ability or end its commitment to make continued payments, so estimates like these are only heuristic guidelines for decision making rather than scientific predictions of actual current values.

Background

2,500 years ago, Greek mathematicians believed that an infinitely long list of positive numbers must sum to infinity. Therefore, Zeno of Elea created a paradox when he demonstrated that in order to walk from one place to another, one must first walk half the distance there, and then half of the remaining distance, and half of that remaining distance, and so on, covering infinitely many intervals before arriving. In doing so, he partitioned a fixed distance into an infinitely long list of halved remaining distances, each with a length greater than zero. Zeno's paradox revealed to the Greeks that their assumption about an infinitely long list of positive numbers needing to add up to infinity was incorrect.

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."

Archimedes' dissection of a parabolic segment into infinitely many triangles

Euclid's Elements has the distinction of being the world's oldest continuously used mathematical textbook, and it includes a demonstration of the sum of finite geometric series in Book IX, Proposition 35, illustrated in an adjacent figure.

Archimedes in his The Quadrature of the Parabola used the sum of a geometric series to compute the area enclosed by a parabola and a straight line. Archimedes' theorem states that the total area under the parabola is 4/3 of the area of the blue triangle. His method was to dissect the area into infinite triangles as shown in the adjacent figure. He 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, then, the total area is the sum of the infinite series $$1+2\left({\frac {1}{8}}\right)+4\left({\frac {1}{8}}\right)^{2}+8\left({\frac {1}{8}}\right)^{3}+\cdots .$$ Here the first term represents the area of the blue triangle, the second term is the area of the two green triangles, the third term is the area of the four yellow triangles, and so on. Simplifying the fractions gives $$1+{\frac {1}{4}}+{\frac {1}{16}}+{\frac {1}{64}}+\cdots ,$$ a geometric series with common ratio $r=1/4$ and its sum is:

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

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.

In addition to his elegantly simple proof of the divergence of the harmonic series, Nicole Oresme proved that the arithmetico-geometric series known as Gabriel's Staircase, $${\frac {1}{2}}+{\frac {2}{4}}+{\frac {3}{8}}+{\frac {4}{16}}+{\frac {5}{32}}+{\frac {6}{64}}+{\frac {7}{128}}+\dots =2.$$ In the 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, representing the geometric series with initial value $a=1/2$ and common ratio $r=1/2$ $$S={\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+{\frac {1}{32}}+\dots ={\frac {\frac {1}{2}}{1-{\frac {1}{2}}}}=1.$$ The second dimension is vertical, where the bottom row is a new initial term $a=S$ and each subsequent row above it shrinks according to the same common ratio $r=1/2$, making another geometric series with sum $T$, $${\begin{aligned}T&=S\left(1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\dots \right)\\&={\frac {S}{1-r}}={\frac {1}{1-{\frac {1}{2}}}}=2.\end{aligned}}$$ This approach generalizes usefully to higher dimensions, and that generalization is described below in § Connection to the power series.