Take the tour ×  I had this problem on my last exam in calculus,
I still don't know, how to solve this, so I would appreciate
your help.  Let $a_{ij} \in \mathbb{R}$ for $i,j \in
\mathbb{N}$. Suppose, that for every $j$ series $S_j =
\sum_{i=1}^{\infty}{a_{ij}}$ is absolutely convergent and, for
every $i$ exists finite limit (limit $\not= \pm\infty$)
$c_i=\lim_{j\to\infty}{a_{ij}}$.  (a) Prove, that if exists such
absolutely convergent series $\sum_{i=1}^{\infty}{b_i}$ that
$|a_{ij}| \leq |b_i|$ for every $i,j$ then $S_j \to S =
\sum_{i=1}^{\infty}{c_i}$, $j \to \infty$  (b) Is argument in
(a) true without assumption that series $\sum{}b_i$ with
properties given in (a) exists?  share|improve this question My
first reaction is that this is the discrete case of the
dominated convergence theorem (i.e., for the counting measure).
This sounds like a bit much for a "calculus" exam though, and a
more elementary explanation would be desirable. Probably someone
else will give one shortly... –  Pete L. Clark Dec 19 '10