# Science:Infinite Series Module/Units/Unit 1/1.3 Infinite Series/1.3.03 The Geometric Series

There are certain forms of infinite series that are frequently encountered in mathematics. The following example

${\displaystyle \sum _{k=1}^{\infty }ar^{k-1}=a+ar+ar^{2}+ar^{3}+\ldots ,\ a\neq 0}$

for constants ${\displaystyle a}$ and ${\displaystyle r}$ is known as the geometric series. The convergence of this series is determined by the constant ${\displaystyle r\in \mathbb {R} }$, which is the common ratio.

Theorem: Convergence of the Geometric Series
Let ${\displaystyle r}$ and ${\displaystyle a}$ be real numbers. Then the geometric series

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

converges if ${\displaystyle |r|<1}$, and otherwise diverges.

## Proof

To prove the above theorem and hence develop an understanding the convergence of this infinite series, we will find an expression for the partial sum, ${\displaystyle s_{n}}$, and determine if the limit as ${\displaystyle n}$ tends to infinity exists. We will further break down our analysis into two cases.

### Case 1: ${\displaystyle r=1}$

If ${\displaystyle r=1}$, then the partial sum ${\displaystyle s_{n}}$ becomes

${\displaystyle s_{n}=\sum _{k=1}^{n}ar^{k-1}=a+a+a+a+\ldots =na.}$

So as

${\displaystyle n\rightarrow \infty }$

we have that

${\displaystyle s_{n}\rightarrow \pm \infty }$.

Hence, the geometric series diverges if r = 1.

### Case 2: ${\displaystyle r\neq 1}$

A short derivation for a compact expression for ${\displaystyle s_{n}}$ will be useful. First note that

align}"): {\displaystyle \begin{align} s_n &= a + ar+ar^2+\ldots+ar^{n-1} \\ rs_n &= ar + ar^2+ar^3+\ldots + ar^{n-1} + ar^n. \end{align}

The second equation is the first equation multiplied by ${\displaystyle r}$. Subtracting these two equations yields

align}"): {\displaystyle \begin{align} s_n - rs_n &= a - ar^{n} \\ s_n &= \frac{a(1 - r^{n})}{1-r}, \ r\ne 1. \end{align}

Using this result, we see that:

• if ${\displaystyle |r|<1}$, then as n → ∞, sna/(1-r)
• if ${\displaystyle |r|>1}$, then as n → ∞, sn → ∞
• if ${\displaystyle |r|=-1}$, then the series is divergent by the Divergence Test (which we cover in a lesson in Unit 2)

## Summary

The above results can be summarized in the following figure.

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