Infinite Series

1. Dec 8, 2003

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?

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

Nille

2. Dec 8, 2003

HallsofIvy

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

3. Dec 8, 2003

himanshu121

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

4. Dec 9, 2003

Derivative86

The answer to the first one is 2.5746952396343726343 Hope that will help

Last edited: Dec 9, 2003
5. Dec 9, 2003

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).

Last edited: Dec 9, 2003
6. Dec 9, 2003

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

7. Dec 13, 2003

laura

This one does not involve hyperbolic trig functions.

Taylor series.

8. Dec 13, 2003

HallsofIvy

Staff Emeritus
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!