I think you are misusing some terminology? You are looking for the sum of 1/n as n goes from 1 to 1000 for your first question, yes? Otherwise I can't make sense of it. (The index of summation here is "n", not 1/n. The 1/n would be called the "summand," I believe.)

Both of these are necessarily finite since we are taking the finite sum of a set of finite numbers. However both of these sums will take a while for your calculator to compute, particularly if you try to do them in "exact" mode.

I get 7.48547 for the first sum and 9.09451 for the second, which shows just how slowly this series diverges.

-Dan