Suppose a series as .

(That is to say has a limit point at infinity.)

Prove that

I have found a number of series where I can prove it case by case, but am having some trouble generalizing. In addition, if the series I am having an embarrassing time showing the result at all.

Another question: Does this theorem have an actual use? I can't think of one.

(FYI: This problem is actually designed for the series to be complex. I figure if I can find out how to prove it for real series then I can generalize it to complex numbers myself.)

Thanks!

-Dan