21 Geometric Series

A Geometric Series on the natural numbers is a series of the form \(\sum\limits_{i=1}^{\infty}a \cdot r ^ {i-1}\) where \(a\neq 0,\ r\neq 0,1\). Expanding the series gives

\[\begin{aligned} \sum\limits_{i=1}^{\infty} a \cdot r^{i-1} &= a \cdot r^0 + a \cdot r ^ 1 + a \cdot r ^ 2 + a \cdot r ^ 3 + \cdots \end{aligned}\]

Alternatively, a Geometric Series may be parameterized on the non-negative integers in the form \(\sum\limits_{i = 0}^{\infty} a \cdot r ^ i\) where \(a\neq 0,\ r\neq 0,1\). Expanding this series gives

\[ \sum\limits_{i=0}^{\infty} a \cdot r^{i} = a \cdot r^0 + a \cdot r ^ 1 + a \cdot r ^ 2 + a \cdot r ^ 3 + \cdots \]

21.1 Partial Summation

21.1.1 Summation for the Series on the Natural Numbers

Let \(S_k\) denote the sum of a series over \(k\) terms (or the \(k^{th}\) partial sum). For the Geometric Series\

\[\begin{aligned} S_k &= \sum\limits_{i=1}^{k} a \cdot r^{i-1} \\ &= a \cdot r^{1 - 1} + a \cdot r^{2 - 1} + a \cdot r^{3 - 1} + a \cdot r^{4 - 1} + \cdots + a \cdot r^{k-1} \\ &= a \cdot r^{0} + a \cdot r^{1} + a \cdot r^{2} + a \cdot r^{3} + \cdots + a \cdot r^{k-1} \\ &= a + a \cdot r + a \cdot r^2 + a \cdot r^3 + \cdots + a \cdot r^{k-1} \\ \end{aligned}\]

Notice that

\[\begin{aligned} r \cdot S_k &= r \cdot \left(a + a \cdot r + a \cdot r^2 + a \cdot r^3 + \cdots + a \cdot r^{k-1} \right) \\ &= a \cdot r + a \cdot r^2+ a \cdot r^3 + \cdots + a \cdot r^k \end{aligned}\]

Taking the difference of these equalities yields:

\[\begin{array}{rrcl} & S_k - r \cdot S_k &=& \left(a + a \cdot r + a \cdot r^2 + a \cdot r^3 + \cdots + a \cdot r^{k-1}\right) - \\ &&& \left(a \cdot r + a \cdot r^2+ a \cdot r^3 + \cdots + a \cdot r^k\right) \\ &&=& a + a \cdot r - a \cdot r + a \cdot r^2 - a \cdot r^2 + \cdots + a \cdot r^{k - 1} - a \cdot r^{k - 1} - a \cdot r ^ k \\ &&=& a + a \cdot r ^ k \\ \Rightarrow & (1 - r) \cdot S_k &=& a - a \cdot r ^ k \\ \Rightarrow & S_k &=& \frac{a - a \cdot r ^ k}{1 - r} \end{array}\]

21.1.2 Summation for the Series on the Non-negative Integers

Let \(S_k\) be the summation of the first \(k\) terms of the geometric series. Whereas the series is on the non-negative integers–that is, the first term is for \(i = 0\), the \(k^{th}\) terms is \(i = k - 1\). This provides

\[\begin{aligned} S_k &= \sum\limits_{i=0}^{k - 1} a \cdot r^{i} \\ &= a \cdot r^{0} + a \cdot r^{1} + a \cdot r^{2} + a \cdot r^{3} + \cdots + a \cdot r^{k-1} \\ &= a + a \cdot r + a \cdot r^2 + a \cdot r^3 + \cdots + a \cdot r^{k-1} \end{aligned}\]

This is identical to the partial summation described in Section 21.1.1, and the derivation of the partial sum is identical.

21.2 Proofs of Convergence

\(\sum\limits_{i=1}^{\infty}ar^{i-1}\) converges when \(|r|<1\) and diverges when \(|r|>1\).

Proof:

Recall that the \(k^{th}\) partial sum of the Geometric Series is

\[ S_k = \frac{a-a \cdot r^k}{1-r} \]

And let \(S\) denote the sum of the infinite series, i.e. the sum as\(k\rightarrow\infty\).

Note that since \(S_k\) is identical whether the series is parameterized on the natural numbers or the non-negative integers, the result is the same for either case.

Case 1: \(|r|<1\)

\[\begin{aligned} S &= \lim\limits_{k\rightarrow\infty}S_k \\ &= \lim\limits_{k\rightarrow\infty} \frac{a-ar^k}{1-r} \\ &= \frac{a-\lim\limits_{k\rightarrow\infty}ar^k}{1-r} \\ &= \frac{a}{1-r} \end{aligned}\]

So \(\sum\limits_{k=1}^{\infty}ar^{k-1}\) converges when \(|r|<1\) and \(S=\frac{a}{1-r}\)

Case 2: \(|r|>1\)

\[\begin{aligned} S &= \lim\limits_{k\rightarrow\infty}S_k \\ &= \lim\limits_{k\rightarrow\infty} \frac{a-ar^k}{1-r} \\ &= \frac{a-\lim\limits_{k\rightarrow\infty}ar^k}{1-r} \\ &= \frac{a-\infty}{1-r} \end{aligned}\]

So \(\sum\limits_{k=1}^{\infty}ar^{k-1}\) diverges whern \(|r|>1\).