Convergence of $ \sum\limits_{n=1}^\infty (a_{n+1} - a_n)$

Question

Let $a_n$ be any convergent sequence of real numbers and let $x_n = a_{n+1} − a_n$ for each $n \in\mathbb{N}$. Prove that the sum $x_n$ as $n$ goes to infinity is a convergent series and find its sum.

Answer 1

This is called a telescopic sum. Try to write explicit expression for the $\sum\limits_{k=1}^N x_k$. You will see that most terms will cancel out. Then take a limit when $N\to \infty$.

Hint

$$ \sum\limits_{k=1}^N x_k = (a_2-a_1)+(a_3-a_2)+(a_4-a_3)+\ldots+(a_{N-2}-a_{N-3})+(a_{N-1}-a_{N-2})= $$ $$ a_2-a_1+a_3-a_2+a_4-a_3+\ldots+a_{N-2}-a_{N-3}+a_{N-1}-a_{N-2}=a_{N}-a_1 $$

We have just computed the partial sum. In order to pass to the infinite series take limit as $N \rightarrow \infty$. $$ \sum\limits_{k=1}^\infty x_k= \lim\limits_{N\to\infty}\left(\sum\limits_{k=1}^N x_k\right)= \lim\limits_{N\to\infty}\left(a_N-a_1\right)= \left(\lim\limits_{N\to\infty}a_N\right)-a_1 $$