Verify that a sum converges to particular function (Fourier Series)

[GamesMasta]

Homework Statement

Verify the formula x=2*(sin(x)-(1/2)sin(2x)+(1/3)sin(3x)-...), {x,-Pi,Pi}

Homework Equations

The Attempt at a Solution

I guess, I am to show that the sum on the right converges to the function x. I began by rewriting the sum on the RHS as $$\displaystyle\sum_{k=1}^k 2*\frac{1}{k}*sin(kx)(-1)^{k-1}$$

Now, I'm not sure what I am to do next. Am I to take the limit of k to infinity? If so how does one solve that? Thank you!

Also, if one graphs this in Mathematica, it can be seen that as k becomes larger and larger the sin function becomes more and more like x through the origin between -Pi and Pi. I believe, however, that I am to show this algebraicly...

 

[Office_Shredder]

Do you know how to calculate Fourier series coefficients? I assume all you're supposed to do is verify that they match what you're given

 

[GamesMasta]

Do you know how to calculate Fourier series coefficients? I assume all you're supposed to do is verify that they match what you're given

Ah, yes I do! I was assuming that since the sum was written that was supposed to use that. I guess if I attempt to calculate the function with only sine terms the coefficients would be given by: $$b_{k}$$=$$\frac{2}{Pi}$$$$\int^{Pi}_{0}f(x)sin(kx)dx$$. You obtain (-1)^(k-1)(2/k) for the coefficient. This then gives you the Fourier Series that gives you the RHS of the original equation. But I am now to prove convergence? That is something I am having difficulty understanding. Any help would be appreciated.

 

[GamesMasta]

Bump.