# What are some examples of convergent series?

Nov 1, 2015

Here are three significant examples...

#### Explanation:

Geometric series

If $\left\mid r \right\mid < 1$ then the sum of the geometric series ${a}_{n} = {r}^{n} {a}_{0}$ is convergent:

${\sum}_{n = 0}^{\infty} \left({r}^{n} {a}_{0}\right) = {a}_{0} / \left(1 - r\right)$

Exponential function

The series defining ${e}^{x}$ is convergent for any value of $x$:

e^x = sum_(n=0)^oo x^n/(n!)

To prove this, for any given $x$, let $N$ be an integer larger than $\left\mid x \right\mid$. Then sum_(n=0)^N x^n/(n!) converges since it is a finite sum and sum_(n=N+1)^oo x^n/(n!) converges since the absolute value of the ratio of successive terms is less than $\frac{\left\mid x \right\mid}{N + 1} < 1$.

Basel problem

The Basel problem, posed in 1644 and solved by Euler in 1734 asked for the value of the sum of reciprocals of squares of positive integers:

${\sum}_{n = 1}^{\infty} \frac{1}{{n}^{2}} = {\pi}^{2} / 6$