What is a convergent sequence and a divergent sequence?

Is the sequence {((-1)^n)/2n} convergent? If so, what is the limit? (I'm thinking that it is convergent)?

• I believe that it would be convergent by the alternating series test, but I am not certain. Can you please tell me if I am on the right track, I would greatly appreciate it! Also, I am somewhat confused on the limit part. Any hints or suggestions for that would also be very appreciated. Thank you!

• Answer:

  You are right, the leibniz test is adequate for the task: it alternates, and 1/(2n) is decreasing. As for the limit, here's some hints. Rewrite as (1/2) Sum( (-1)^n /n ). Then think of the series it could come from: Sum( (-1)^n x^n /n ), for x=1. This is the series for which function? To see it, call f(x) the above sum, and find its derivative; the resulting series should be recognizable, and its primitive also. Then evaluate at x=1.