How to estimate infinite sums

How would you do this problem?

sum from k = 1 to infinity, [(-1)^k] / (2k+1)^3

This is an alternating series; so the error after n terms is bounded above by the absolute value of the following (n+1)-th term.

For instance, if we need the error less than 1/1000, we require that
1/(2(k+1) + 1)^3 < 1/1000
==> (2k+3)^3 > 1000
==> 2k + 3 > 10
==> k > 7/2.

So, we need at least 4 terms for the desired accuracy.

I hope this helps!