Question

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

Answer 1

The series converges absolutely.

Explanation:

First note that:

`(4+abs(cosk))/k^3<=5/k^3` for `k=1...oo`

and

`(4+abs(cosk))/k^3>0` for `k=1...oo`

Therefore if `sum5/k^3` converges so will `sum(4+abs(cosk))/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:

See http://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/SandS/SeriesTests/p-series.html for more info.