1)If I have something like sum (j=1 to inf) 1/j and want to estimate the first 10^10 terms, how would I do that?

2) How large does N have to be so sum (n=1 to N)1/n^5 is within 10^-6 of sum (n =1 to inf) 1/n^5

I just set integrated 1/n^5 from 1 to N and set it to 10^-6 and solved for N. Is this right?