Fourier series homework problem

[Bohrok]

Not really a homework question; I typed this sum into Wolframalpha and it gave a nice, compact expression, but I couldn't figure out where to begin finding it. Is there a way to find it using just calc II-level knowledge of infinite sums?

[PLAIN]http://www4c.wolframalpha.com/Calculate/MSP/MSP839119dd21ei6e3ic70g00005i4356075068c2eb?MSPStoreType=image/gif&s=40&w=264&h=45

 

[╔(σ_σ)╝]

Yes. By using a Fourier series.

 

[Bohrok]

Fourier series are beyond anything I learned in calc II... Is that the only way to do it?

 

[Char. Limit]

Fourier series are beyond anything I learned in calc II... Is that the only way to do it?

In many cases, the sum can only be obtained by complicated methods like the Fourier Series. In far more cases, the sum can only be approximated. Personally, I would be happy that this one falls in the first category... but I can see why you would want to be able to do this. I have finished Calc III, and I still don't see a way to do this.

 

[Telemachus]

The method that you can use which requires only calc I knowledge is using improper integrals for approximating the sum. I saw an example of this on the Stewart (Calc I book which I'm using for my final exam).

 

[Char. Limit]

The method that you can use which requires only calc I knowledge is using improper integrals for approximating the sum. I saw an example of this on the Stewart (Calc I book which I'm using for my final exam).

Yes, but that only gives him an approximation. I believe he's looking for a way to retrieve the closed-form solution.

 

[Bohrok]

Yes, I was looking for a way to find the expression \frac{1}{2}(1 + \pi \coth(\pi)) that Wolframalpha gave for the sum; I kinda figured I couldn't do it yet with that coth in there. I'll just have to wait till Fourier series...
Thanks Char