# some questions on convergence of sequences

• Jan 11th 2012, 02:16 AM
alphabeta89
some questions on convergence of sequences
Let $\displaystyle (a_n)$ be a sequence.

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

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

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

• Jan 11th 2012, 04:50 AM
CaptainBlack
Re: some questions on convergence of sequences
Quote:

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

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

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

CB
• Jan 11th 2012, 06:25 AM
alphabeta89
Re: some questions on convergence of sequences
Quote:

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

CB

hmm? I think it still works for this sequence.
• Jan 11th 2012, 04:33 PM
alphabeta89
Re: some questions on convergence of sequences
Anyone has any idea on starting this question?