A question concerning series of functions.

Question

Suppose $\sum_1^\infty a_n$ converges. Then is it true that $\sum_1^\infty b_n$ is convergent and $b_n$ goes to zero as $n\to \infty$? Here $b_n =a_{n+1}-a_n$. I trued for some counter examples but did not get any. Thanks in advance.

Answer 1

Your partial sums will be telescopic, that is $b_1+b_2+\dots+b_n=(a_2-a_1)+(a_3-a_2)+(a_4-a_3)+\dots+(a_{n+1}-a_n)=a_{n+1}-a_1$ (Observe how the intermediate terms are cancelling each other). Now since $\sum_na_n$ converges, $a_n\to 0$, therefore $b_1+\dots+b_n=a_{n+1}-a_1\to -a_1$.

Answer 2

Same as @JustDroppedIn's aproach, but set out differently, $$S=\sum_{n=1}^\infty b_n=\sum_{n=1}^\infty a_{n+1}-a_n=(a_2-a_1)+(a_3-a_2)+(a_4-a_3)+\cdots=-a_1.$$ Since $a_1$ is bounded, so is $S$.