Hey MSUMathStdnt.
Have you tried using some other tests like the integral test?
A Theorem say the -series converges if and diverges otherwise.
The question says "Find the positive values of for which the series converges.
The answer is (from back of book). There is no explanation of how they got it and I have none either. How do you find that?
I thought maybe I would compare to . If , then the former converges, too. For , and since , then and thus . How am I doing so far?
But what about when ? Maybe for some values close to, but less than 1, still converges. How do I find out?
First a correction. The lower bounds was supposed to be , not . If you try to take the in the denominator... well, you know.
OK, so here's the integral test (so far)
The first term
for all positive values of .
The second term
for all .
So the entire integral comes to , which means that the series converges when that fraction converges. Right so far? If so, now what?