# Almost zeta(2)

1. May 8, 2013

### Office_Shredder

Staff Emeritus
I'm noticing wolfram alpha has the amazing ability to analytically solve
$$\sum_{n=1}^{\infty} \frac{1}{n^2 + a^2}$$

Anyone know how to do this, and if it's also possible to deal with higher order guys (like it also can do 1/(n4+a2), but it's a way more complicated expression to the point where I'm staring at it wondering if it's actually a real number)

2. May 8, 2013

### mathman

It is a real number. However I have no idea how to calculate it.

3. May 8, 2013

### Mandelbroth

I'd split it into partial sums and then evaluate.

Hint:
$\frac{1}{n^2+a^2}=\frac{1}{(n+ai)(n-ai)}$, where i is the imaginary unit.

Higher order expressions will undoubtedly be a lot more complicated. Though, given n and a are real numbers, the sum will also be real, so they won't necessarily be more complex (heh. See what I did there? :tongue:).

4. May 8, 2013

### Office_Shredder

Staff Emeritus
If you split it into two fractions of degree one neither of your series converge anymore.

By "wondering if it's a real number" I meant "gee wolfram alpha has a lot of 4th roots of -1 in that expression" not "I literally don't know if it's a real number"

5. May 9, 2013

6. May 9, 2013

### Office_Shredder

Staff Emeritus
Oh wow that was easy. I was too busy approaching the problem from the wrong side of the equation. Thanks

7. May 9, 2013

### dextercioby

It is actually not easy, not to me. I can't find a proof for the series expansion. :D