What are some examples of convergent series?

1 Answer
Nov 1, 2015

Answer:

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#