TITLE: When does l'Hospital's rule work for series?
QUESTION [19 upvotes]: This question stems from a response a colleague of mine received as a (flawed) solution to a problem on his calculus exam. The student was to determine convergence of a series of the form $\displaystyle\sum_{n=1}^\infty \frac{f(n)}{g(n)}$. The student instead considered the series  $\displaystyle\sum_{n=1}^\infty \frac{f'(n)}{g'(n)}$ and thought that this series and the original must behave the same way. This, of course, is false in general.

What conditions on $f$ and $g$ are required so that the series  $\displaystyle\sum_{n=1}^\infty \frac{f(n)}{g(n)}$ converges iff  $\displaystyle\sum_{n=1}^\infty \frac{f'(n)}{g'(n)}$ converges?

REPLY [3 votes]: I'm the guy who had the confused student.
In my original posing of the question, I ask for $f'$ and $g'$ to be monotonic, not just $f$ and $g$.  Are you still able to create a counterexample?  
Someone mentioned usefulness.  If something like this worked, I'd have students apply it to series like n/(n^3-1).  The Comparison Test is awkward to use correctly.  The Limit Comparison Test is great for this one, but it's nice to have options.
Maybe the list of conditions to make this conjecture true becomes too lengthy to bother with.  But I have yet to see a counterexample to these conditions:

$f'$ and $g'$ are monotonic.
$\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=0$
$f$ and $g$ have "L'Hopital conditions": $\lim_{x\rightarrow\infty}f(x)=\lim_{x\rightarrow\infty}g(x)=0$ or $\pm\infty$.