Summation of a Product of Functions

[drewfstr314]

Is there a general formula for something like

$\sum_{n=0}^{\infty} \left( f(n) \times g(n) \right)$



For example, what is

$\sum_{n=0}^{\infty} \left( 3^n \times \frac{n!}{n^2} \right)$

 

[micromass]

For example, what is

$\sum_{n=0}^{\infty} \left( 3^n \times \frac{n!}{n^2} \right)$

Infinity, since the series diverges.

 

[Stephen Tashi]

In special cases, you can make progress by using "summation by parts". (http://en.wikipedia.org/wiki/Summation_by_parts)