The Geometric Series

Previous: Convergence of Infinite Series

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

$\text{[math]}$

for constants $\text{[math]}$ and $\text{[math]}$ is known as the geometric series. The convergence of this series is determined by the constant $\text{[math]}$ , which is the common ratio.

Theorem: Convergence of the Geometric Series
Let $\text{[math]}$ and $\text{[math]}$ be real numbers. Then the geometric series $\text{[math]}$ converges if $\text{[math]}$ , and otherwise diverges.

Theorem: Convergence of the Geometric Series

Let

\text{[math]}

and

\text{[math]}

be real numbers. Then the geometric series

$\text{[math]}$

converges if $\text{[math]}$ , 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, $\text{[math]}$ , and determine if the limit as $\text{[math]}$ tends to infinity exists. We will further break down our analysis into two cases.

Case 1: $\text{[math]}$

If $\text{[math]}$ , then the partial sum $\text{[math]}$ becomes

$\text{[math]}$

So as

$\text{[math]}$

we have that

$\text{[math]}$ .

Hence, the geometric series diverges if r = 1.

Case 2: $\text{[math]}$

A short derivation for a compact expression for $\text{[math]}$ will be useful. First note that

$\text{[math]}$

The second equation is the first equation multiplied by $\text{[math]}$ . Subtracting these two equations yields

$\text{[math]}$

Using this result, we see that:

if $\text{[math]}$ , then as n → ∞, s_n → a/(1-r)
if $\text{[math]}$ , then as n → ∞, s_n → ∞
if $\text{[math]}$ , then the series is divergent by the Divergence Test (which we cover in a lesson in Unit 2)