A geometric series is a series where each term is a constant multiple of the last term. This constant multiple is known as the common ratio, and is denoted by $r$.
Here’s a classic example:
$$ 1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots $$
Here, each term is half of the previous term, so the common ratio is $\frac{1}{2}$.
The general form of a geometric series with first term 1 and common ratio $r$is:
$$ \sum_{i=0}^\infty r^i = 1 + r + r^2 + r^3 + \cdots $$
To find a general formula for the sum of geometric series, we first need to consider the sum of the first $n$ terms of one.
Consider the sum $\displaystyle\sum_{i=0}^{\infty} = 1+r+r^2+r^3+\cdots$. The sum of the first $n$ terms is:
$$ s_n = 1+r+r^2+\cdots+r^{n-1}+r^n $$
Therefore:
$$ rs_n = r+r^2+r^3+\cdots+r^n+r^{n+1} $$
Subtracting the second equation from the first, we get:
$$ s_n-rs_n = (1 - r) + (r - r^2) + (r^2-r^3) + \cdots + (r^{n-1}-r^n) + (r^n - r^{n+1}) $$
$$ s_n(1-r) = 1-r^{n+1} $$
$$ s_n = \frac{1-r^{n+1}}{1-r} $$
Now we take the limit as $n$ approaches infinity to get a formula for the sum of the whole series.
$$ \lim_{n\to\infty}s_n = \lim_{n\to\infty}\frac{1-r^{n+1}}{1-r} $$
There are 3 cases: