# How do you test for convergence for sum(4 + abs(cosk))/(k^3) for k=1 to infinity?

Aug 3, 2015

The series converges absolutely.

#### Explanation:

First note that:

$\frac{4 + \left\mid \cos k \right\mid}{k} ^ 3 \le \frac{5}{k} ^ 3$ for $k = 1. . . \infty$

and

$\frac{4 + \left\mid \cos k \right\mid}{k} ^ 3 > 0$ for $k = 1. . . \infty$

Therefore if $\sum \frac{5}{k} ^ 3$ converges so will $\sum \frac{4 + \left\mid \cos k \right\mid}{k} ^ 3$ since it will be less than the new expression (and positive).

This is a p series with $p = 3 > 1$.

Therefore the series converges absolutely: