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

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Answer 1 · 2015-08-03T08:40:51

The series converges absolutely.

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: