Question

For the following series, does it diverge or converge? If it converges, what is the limit?
Sum from n=1 to inf of ln(2(n+1))-ln(2n)

Answer 1

This seems to diverge. I would break up into two sums:\[\sum_{1}^{\infty} \ln(2(n+1)) - \sum_{1}^{\infty} \ln(2n)\]Remember that if we have two sums, if one diverges, the sum diverges. So, we have to prove that either sum diverge (both diverge, pick one and solve it).

Answer 2

Typo: I meant, if we have a sum of two sums, if one of the sums diverges, the sum of sums diverges, i.e.,\[\sum_{i = 1}^{\infty} a_{i} + \sum_{i=1}^{\infty} b_i\]if {a_i} or {b_i} diverges, their sum diverge.

Answer 3

Thank you.