I've been having some difficulty solving this question especially with understanding how to comput the specific subesequence specified and why they relate to solving this question.
The question is:
Using the steps below, show that the following sequence converges:
1 + 1/2 - 2/3 + 1/4 + 1/5 - 2/6 + 1/7 + 1/8 - 2/9 + 1/10 + 1/11 - 2/12 + + - + +...
Consider the subsequence (s2, s3, s5, s6, s8, s9, ...) of the sequence of partial sums. Show that this is the sequence of partial sums of a related convergent series.
So far I have considered a sequence (sn) of the partial sums such that
(sn) = ∑(1/3n+1 + 1/3n+2 - 2/3n+3) = 2/(3n+1)(3n+3) + 1/(3n+2)(3n+3) ~ 1/3n^2
but I am not sure of how to compute the specified subsequences from this.