If each term in a sum converges, does the infinite sum converge too? Let $S(x) = \sum_{n=1}^\infty s_n(x)$ where the real valued terms satisfy
$s_n(x) \to s_n$ as $x \to \infty$ for each $n$. 
Suppose that $S=\sum_{n=1}^\infty s_n< \infty$.
Does it follow that $S(x) \to S$ as $x \to \infty$?
I really do not recall any theorems... please point out !!!

No.
Let $s_n(x)=\frac 1x$, $s_n=0$. Then $S=\sum_{n=1}^\infty s_n$ trivially converges. However, $S(x)=\sum_{n=1}^\infty s_n(x)$ is clearly divergent, since the summand is a constant (with respect to $n$) greater than $0$.
