Prove F(n)→0 as n→∞ Warning: Danger Ahead?

[phoenixthoth]

This has been a cool problem, if damn frustrating.

I bought some small books on asymptotic analysis because this stuff is really interesting.

Thanks for posting it.

Cheers

 

[steven187]

hello all

in terms of this problem I have been looking at the laurent series, I have been playing around with it but I keep coming across dead ends, see I remember reading once that you can do a lot with the complex field especially to solve problems in the real number field, now to find the sum of a series, would using the laurent series be the best place to start for attempting this problem in the complex field? if not where is the best place to start to find the sum of a series in the complex field? any suggestions would be helpful

steven

 

[steven187]

hello all

well even after doing some research into complex analysis it didnt really give me much help, the only thing that i could find that could be possibly related to it is the laurent series but can't figure out how to apply it to this problem, so I decided to go and as a friend who is a lecturer in analysis, he had one look at this question and said it definitely has something to do with this theorem how sounded very certain

Cauchys Theorem
if f(z) is analytic and

$$\frac{f(z)}{z-z_{o}}$$

has a simple pole at $$z_{0}$$

with residue $$f(z_{o})$$

then the theorem says that if f(z) is analytic within C the value of f at some point $$z_{0}$$
within C is given by

$$f(z_{0})=\frac{1}{2\pi i} \oint_{C}\frac{f(z)}{z-z_{o}} dz$$

would anybody have any idea on how to apply this theorem to this Problem, I honestly can't see the link, any suggestions would be appreciated

steven