Infinite Series

[nille40]

Hi all!
I was wondering which method one should use to find the actual sum of an infinite series. I know how to find the sum of a geometric series (if it converges), but how could I find the sum for, for instance

\sum_{n=0}^\infty\left(\frac{n+5}{5n+1}\right)^n


I know that it converges, and the sum appears to be 2. But how can I calculate this?

Or how about

\sum_{n=0}^\infty\frac{\cos n\pi}{1+n^2}


Thanks in advance,
Nille

[HallsofIvy]

There is no general method for determining the sum of an infinite series.

[himanshu121]

What if we try to find the sum to n terms and then taking lim n\rightarrow\infty

[Derivative86]

The answer to the first one is 2.5746952396343726343 [a)] Hope that will help

[suyver]

The second one is a defined convergent series:

\sum_{n=0}^\infty\frac{\cos n\pi}{1+n^2}=\frac12\left(1+\pi{\rm csch}(\pi)\right)

where csch(z) gives the hyperbolic cosecant of z, or in other words: csch(z)=1/sinh(z).

[himanshu121]

How You guys reach this conclusions I have read a little about convergence and divergence but don't know how you summed up the series

[laura]

Originally posted by nille40

\sum_{n=0}^\infty\frac{\cos n\pi}{1+n^2}


This one does not involve hyperbolic trig functions.

Taylor series.

[HallsofIvy]

laura: would you mind explaining further? Since the sum, as written, is clearly not a Taylor series, do you mean that it can be converted to one and then summed? If so, how? It's certainly not obvious to me!