Definition

A series is called geometric if it is of the form $$ \begin{align*} \sum_{k=1}^{\infty} ar^k = a + ar + ar^2 + ar^3 + ... \end{align*} $$

The Partial Sum Sequence

To know anything about whether this series converges or diverges, we look at its partial sum sequence $(s_m)$ where $s_m$ is

$$ \begin{align*} s_m = a + ar + ar^2 + ... + ar^{m-1}. \end{align*} $$

When $r \neq 1$, we can use the following algebraic identity:

$$ \begin{align*} (1 - r)(1 + r+ r^2 + r^3 + ... + r^{m-1}) = 1 - r^m. \end{align*} $$

to simplify the partial sum sequence into

$$ \begin{align*} s_m &= a + ar + ar^2 + ar^3 + ... + ar^{m-1} \\ &= a(1 + r + r^2 + r^3 + ... + r^{m-1})\\ &= \frac{a(1 - r^m)}{1 - r} \end{align*} $$

Convergence

When $r = 1$ and $a \neq 0$, we have

$$ \begin{align*} \sum_{k=1}^{\infty} ar^k = \sum_{k=1}^{\infty} a. \end{align*} $$

This sum diverges. It will never converge to anything. We can in fact generalize this for any $|r| \geq 1$. The sum will just diverge. However, when $|r| < 1$, we know from earlier that the sequence of partial sums $(s_m)$ can be written as

$$ \begin{align*} s_m = \frac{a(1 - r^m)}{1 - r} \end{align*} $$

We know $(r^m)$ convergences to zero (proof here). $\frac{1}{1-r}$ is a constant so we can use the Algebraic Limit Theorem for Sequences to conclude that

$$ \begin{align*} \lim s_m = \frac{a}{1 - r}. \blacksquare \end{align*} $$

Other Definitions and Properties

Fo the absolute value function definition and other properties, see here.
For the definitions of sequences, subsequences and what it means to for a sequence to converge, see this.
For the definitions of series, and what it means to for a series to converge, see this.
For the "show the limit" template and an example, see this.

References: