# Thread: some questions on convergence of sequences

1. ## some questions on convergence of sequences

Let $(a_n)$ be a sequence.

(i) Prove that if $\sum_{n=1}^{\infty}{a_n}$ converges, then $\sum_{n=1}^{\infty}{(a_{2n-1}+a_{n})}$ also converges.

(ii) Prove that if $\sum_{n=1}^{\infty}{(a_{2n-1}+a_{n})}$ converges and $a_n \rightarrow {0}$, then $\sum_{n=1}^{\infty}{a_n}$ converges.

For (i), can i use the concept of subsequences to tackle this question.
For (ii), how do I start this question?

2. ## Re: some questions on convergence of sequences

Originally Posted by alphabeta89
Let $(a_n)$ be a sequence.

(i) Prove that if $\sum_{n=1}^{\infty}{a_n}$ converges, then $\sum_{n=1}^{\infty}{(a_{2n-1}+a_{n})}$ also converges.
I'm not sure this is true. What happens if $a_n=(-1)^{n-1}/n$, then $\sum_{n=1}^{\infty}{a_n}$ converges but I don't think $\sum_{n=1}^{\infty}{(a_{2n-1}+a_{n})}$ does.

CB

3. ## Re: some questions on convergence of sequences

Originally Posted by CaptainBlack
I'm not sure this is true. What happens if $a_n=(-1)^{n-1}/n$, then $\sum_{n=1}^{\infty}{a_n}$ converges but I don't think $\sum_{n=1}^{\infty}{(a_{2n-1}+a_{n})}$ does.

CB
hmm? I think it still works for this sequence.

4. ## Re: some questions on convergence of sequences

Anyone has any idea on starting this question?