# Sufficient criterion for convergence of series

Let $$(a_k)$$ be a sequence decreasing to $$0$$, for which $$b_k:=\left(\sum_{l=1}^ka_l\right)-ka_{k+1}=\sum_{l=1}^k(a_l-a_{k+1})$$ is bounded. Does $$\sum_{k\geq 0}a_k$$ necessarily converge?

Some immediate observations:

• $$(b_k)$$ is increasing, hence converges.
• By considering $$b_k-b_{k-1}$$, one obtains that $$k(a_k-a_{k+1})$$ tends towards $$0$$.
• For $$\sum a_k$$ to converge, it suffices to prove that $$ka_k$$ converges.

We know that $$(b_k)_{k\in\mathbb N}$$ is convergent. Let $$b$$ be its limit.

Let $$p$$ be a positive integer.

$$\forall k \geqslant p \quad , \quad b_k = \displaystyle \sum_{\ell = 1}^k (a_{\ell}-a_{k+1})\geqslant \sum_{\ell =1}^p (a_{\ell}-a_{k+1})$$

Then $$\quad \lim b_k \displaystyle \geqslant \lim_{k\rightarrow +\infty} \sum_{\ell=1}^p (a_{\ell}-a_{k+1})$$

So $$\quad b \geqslant \displaystyle\sum_{\ell = 1}^p a_{\ell}$$

And we can conclude that $$\sum a_k$$ is convergent.

• Very nice. Thanks a lot! – Zuy Mar 14 at 5:53