Finding the value a series converges to

[thomas49th]

I have

$$\sum^{\infty}_{n=1}\frac{(-1)^{n+1}}{n^{2}}$$



I need to show it equals pi^2 / 12. Not sure where to begin :\ Tried plugging in values but non cancel out which I can see :\

 

[LCKurtz]

Does this problem arise for you in a Fourier Series course, or if not that, where?

 

[thomas49th]

yes, after computing the Fourier series I am then asked to find this (changed slightly though as before it wasn't n+1... just n). How did you guess btw and why should it matter?

 

[LCKurtz]

yes, after computing the Fourier series I am then asked to find this (changed slightly though as before it wasn't n+1... just n). How did you guess btw and why should it matter?

How did I guess? Well, given that it has a pi in the answer, it isn't likely you are going to find the sum with simple algebra; it is going to take something more advanced. And this problem is the type typically asked in Fourier Series chapters where you are learning about how the FS converges.

You didn't state what FS you calculated, but if you evaluate it at the magic point and use what you know about what the series converges to at continuity points or finite jumps (whichever is relevant to your problem), the result will drop out.

 

[thomas49th]

the magic point? How do I find that?

 

[LCKurtz]

the magic point? How do I find that?

You say "presto - change-o" and try the simplest point you can think of. Be brave...

 

[thomas49th]

I shall try n=1. Therefore -1 pops out. What do I know about series convergence. Not allow. Some series convergences to a single value. I recall something about continuous and jumpy series. If your series jumps around you can get Gibbs Phenomena. Lovely stuff but don't think that is what you were after? Also in discontinuous series the Fourier series always goes through the middle point or average of the discontinuity between each peroid.

Am I any closer?

Thomas

 

[LCKurtz]

I shall try n=1. Therefore -1 pops out. What do I know about series convergence. Not allow. Some series convergences to a single value. I recall something about continuous and jumpy series. If your series jumps around you can get Gibbs Phenomena. Lovely stuff but don't think that is what you were after? Also in discontinuous series the Fourier series always goes through the middle point or average of the discontinuity between each peroid.

Am I any closer?

Thomas

No, no... Look, you have a function f(x) and you have expanded it in a FS, which I will call s(x)

So you have f(x) = s(x), hopefully, for most values of x (you have a theorem about that). Put the simplest x you can think of in both sides of that equation and see what happens.

 

[thomas49th]

very nice

set t = 0

as adding 1 to the exponent of (-1)^n is the same as multiplying by -1 we find that we get pi^2 / 12

THANKS!

 

[dextercioby]

Can you find the following value ?

$$\sum_{n=1}^{\infty} \frac{1}{n^2}$$

If you do, then what you have to find is half of it.

 

[LCKurtz]

very nice

set t = 0

as adding 1 to the exponent of (-1)^n is the same as multiplying by -1 we find that we get pi^2 / 12

THANKS!

You're welcome. Glad you found the magic point.

 

[dextercioby]

Can you find the following value ?

$$\sum_{n=1}^{\infty} \frac{1}{n^2}$$

If you do, then what you have to find is half of it.

On a second thought, there's a method of finding exactly the sum you're looking for

Assume that y=x^2 is defined on the interval (-pi, pi). Find the Fourier series for it and then plug x=0. You'll get exactly the formula you're looking for.