Consider the series

Determine whether the series converges or diverges. If it converges, determine whether it converges conditionally or absolutely.

The series converges.

*Proof.* Let . Then, the series converges. Using the limit comparison test we have

Hence, and both converge or both diverge. Since converges we have established the convergence of

The integral goes to infinite, not 1. lim n-> infinite of sin(1/n) / (1/n) goes to infinite because 1/n goes to 0 and since it is in the denominator the whole expression goes to infinite

You should review the basic limit facts