Dirichlet's Test

Let

$\begin{displaymath} \left\vert{\,\sum_{n=1}^p a_n}\right\vert < K, \end{displaymath}$

where

is independent of

. Then if $f_n \geq f_{n+1} > 0$ and

$\begin{displaymath} \lim_{n\to\infty} f_n=0, \end{displaymath}$

it follows that

$\begin{displaymath} \sum_{n=1}^\infty a_n f_n \end{displaymath}$