Series

The sum of an infinite geometric progression.

How to solve it

Geometric series

The infinite sum $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$ equals a finite number, $2$: each new term covers half of the remaining way. Every geometric progression whose ratio is less than one in absolute value behaves like this.

Step by step

Find the first term $b_1$, the term the sum starts with. Look carefully at the lower index of the sum: if it starts at $k = 2$, the first term is the one with $k = 2$.
Find the ratio $q$: how many times each term is larger than the previous one.
Check that $|q| < 1$. Otherwise the series diverges and has no sum.
Compute the sum by the formula.

The first term of the sum.

The ratio of the progression, $|q| < 1$.

Example: $\sum_{k=0}^{\infty} \left(\frac{1}{2}\right)^k = \frac{1}{1 - \frac{1}{2}} = 2$.

Common mistakes

Taking the term at $k = 0$ as the first one, although the sum starts at $k = 2$.
Getting the sign wrong for a negative ratio: $1 - \left(-\frac{1}{2}\right) = \frac{3}{2}$.
Using the formula when $|q| \ge 1$ and the series diverges.