Minimum number of terms to approximate the series within .001

sum 1/n^3 n=1..infinity I have gotten as far as getting through the integration. I cannot get beyond 1/2n^2 <0.001

I know the answer should be 23 terms, but I can't figure out how to get there (I guess the problem that I am having is figuring out how to get n^2 by itself) . Thanks