Let $(a_n)_{n\geq1}$ be a nondecreasing sequence of positive numbers converging to some limit $l$.
Show that $\sum_{k=1}^{\infty}(l-a_k)$ converges if and only if $\lim_{n\to\infty}\frac{a_1\cdot a_2\cdot …\cdot a_n}{l^n}=C$ for some $C>0$.
My thoughts:
In the "forward" direction, Given $\varepsilon>0$ there exists an $N$ such that whenever $m > n\geq N$,
$|\sum_{k=n+1}^{m}(1-\frac{a_k}{l})| < \frac{\varepsilon}{l}$ by the Cauchy Criterion.
We also know that $\lim\frac{a_n}{l}=1$ with $0<\frac{a_n}{l}<1$ for all $n$.
For finite $N$ we know that $(a_1a_2…a_N)/l^N$ is also positive and finite.
If anyone could offer any hints that'd be appreciated. Thanks.
Best Answer
Note that $l\geq a_1>0$. Let $b_k=1-\frac {a_k} l$. The statement now becomes the following:
$\sum b_k$ converges if and only if $\prod (1-b_k)$ converges to a positive number. This equivalence is a standard fact about infinite products. It is proved by going to logarithms and using that fact that $\ln (1+x) \sim x$ as $x \to 0$. Ref: Rudin's RCA.