Question

What are some examples of convergent series?

Answer 1

Here are three significant examples...

Explanation:

Geometric series

If `abs(r) < 1` then the sum of the geometric series `a_n = r^n a_0` is convergent:

`sum_(n=0)^oo (r^n a_0) = a_0/(1-r)`

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 `abs(x)`. 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 `abs(x)/(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)^oo 1/(n^2) = pi^2/6`