calculus - Prove or disprove the converse of a proposition of test of convergence of series

We can see the fact that:

If a series $\sum_{n=1}^{\infty} a_{n}$ converges then:

$\displaystyle \lim_{n \rightarrow \infty} (a_n + a_{n+1} +···+ a_{n+r} )=0 $

This is my proof:

$\displaystyle \lim_{n \rightarrow \infty} (a_n + a_{n+1} +···+ a_{n+r} )$

$=\displaystyle \lim_{n \rightarrow \infty} a_n + \displaystyle \lim_{n \rightarrow \infty}a_{n+1} +···+\displaystyle \lim_{n \rightarrow \infty} a_{n+r} $

$=0+0+...+0=0$

Is it correct?

Also I want to ask:Does the converse of the implication holds:

That it: Does $\displaystyle \lim_{n \rightarrow \infty} (a_n + a_{n+1} +···+ a_{n+r} )=0 $
imply the series $\sum_{n=1}^{\infty} a_{n}$ convergent?

Whether it is true or not. I am searching for a proof and a justification. Could someone help to prove or disprove the statement?

Thanks so much !

Answer

As long as $r$ is finite, I believe your answer is correct. The converse is not true. Let's, for example, let $a_n = 1/n.$