Suppose $\{a_n\}$ is a sequence of positive terms.
It is a well known result that if the series $\displaystyle \sum_{k=1}^{\infty} a_k$ converges then $\displaystyle \lim_{m \rightarrow \infty} \displaystyle \sum_{k=m}^{\infty} a_k=0 $.
Is the converse true?
That is, is it true that if the tail of a series goes to zero, then the series must converge?
My thoughts:
If we let $S_n=\displaystyle \sum_{k=1}^{n} a_k$, then clearly $S_n$ is an increasing sequence, and for $n > m$ , we have $S_n – S_m = \displaystyle \sum_{k=m+1}^{n} a_k $
Then we want to show that if $\displaystyle \lim_{m\rightarrow \infty} (\displaystyle \lim_{n\rightarrow \infty} S_n – S_m ) = 0$ (or just if it exists) then $\displaystyle \lim_{n\rightarrow \infty} S_n < \infty$.
Note that since $S_n$ is increasing, it is enough to show that it is bounded.
I tried to show this by definition, but my issue is that I am not sure how to deal with that double limit.
Any help would be really appreciate it.
Thanks!
Best Answer
The result you want is obvious. If $$\lim_{m\to\infty } \sum_{n=m}^{\infty} a_n = 0$$ then for some $M$ we have that $$\sum_{n=M}^{\infty} a_n <\infty.$$ Tacking on the first $M$ terms doesn't affect convergence.