Are there rules for answering this question:
$f(x)$ is a smooth function that's always increasing without bound, and $g(x)$ is a smooth function that always decreases from 1, approaching zero but never equaling it. $\sum_{x=1}^nf(x)$ diverges as n approaches infinity. But we know for sure that $\sum_{x=1}^ng(x)$ converges as n approaches infinity. Will $\sum_{x=1}^nf(x)\cdot g(x)$ converge or diverge? $f(x)$ and $g(x)$ are positive for all $x\ge 1$.