alternating series test.
To determine if converges, set .
You can see that decreases as increases. I'll demonstrate by showing if it were a function on the reals, its derivative would be negative on the relevant domain:
, which is negative for , which is where we're counting from.
The alternating series test also requires that . This is true and easy to verify using L'Hospital's rule.
The conditions of the alternating series test are satisfied. The series converges.