How do you prove divergent series? (sum of series is infinite?)?

Question

If the series was:
(1/1) + (1/2) + (1/3) + (1/4) + (1/5) + .. + (1/infinite),
is it true that the sum of this series is infinite? If so, how do you prove this?

Answer 1

one way is to compare it to the integral of its corresponding function. The function in your example is 1/x.
If we take the improper integral of x from 0 to infinity we find that the integral goes to infinite and thus does not converge. If the integral does not converge, the series does not converge. If the integral had converged, then the series would have converged.

Answer 2

Take the limit of the integral as the series approaches infinity. Basically that series is 1/n. The integral of 1/n is ln(n). Ln(infinity)=infinity. However if it was 1/n^2 it would be -1/2*1/n, whose limit as it approaches infinity would be 0.