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.
Convergence of $ \sum\limits_{n=1}^\infty (a_{n+1} - a_n)$
2
$\begingroup$
real-analysis
sequences-and-series
-
0@Norbert: I am not sure whether it is good to use `\limits` in the title - for the reasons similar to: [Why no use displaystyle in titles?](http://meta.math.stackexchange.com/questions/3135/why-no-use-displaystyle-in-titles/). – 2012-07-11
-
0@MartinSleziak, \limits often used in question om MSE, and for my taste I don't like \sum signs without specification range of summing index. If you don't agree you can rollback or edit it in an appropriate way – 2012-07-11
-
0@Norbert You're right, using \limits in the titles seems to be ok. (I remembered this was discussed somewhere and found [this](http://chat.stackexchange.com/transcript/message/2709198#2709198).) – 2012-07-11