Calculating Sums of Infinite Series for Nille

[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 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!