Prove that $(a_1-a_2)+(a_2-a_3)+....$ converges iff ${a_n}$ converges Let $b_n=a_n-a_{n+1}$.
We first assume that ${a_n}$ converges so ,$\lim(b_n)=\lim(a_n)-\lim(a_{n+1})$ , hence $\lim(b_n)=0$.
Now let $s_m$ and $s_n$ be the consecutive partial sums of ${b_n}$.
So $|s_n-s_m| = |a_{n+1}-a_{m}|=|a_{n+1}-L+L-a_m|< \epsilon$ for $N\ge M \ge M(\epsilon)$.So the series is cauchy convergent .
Now let us assume that the series is convergent so the sequence formed by the partial sums are convergent hence $s_M=a_1-a_{M+1}$ hence $-\lim(s_M) +a_1 = \lim(a_{M+1})$.So $A=-S+a_1$.
Can someone go through my attempt and point out my mistake instead of suggesting another answer. 

It's a telescoping series.
$\sum_{k=1}^n (a_k - a_{k+1}) = a_1 - a_{n+1}$.
so $\lim\limits_{n\to \infty} \sum_{k=1}^n (a_k - a_{k-1}) = \lim\limits_{n\to \infty} (a_1 - a_{n+1}) = a_1 - \lim\limits_{n\to \infty} a_n$
Which converges and exists if and only if the $a_n$ converges.
