Hi! I need to show that
for .
This looks like the sort of sum that arises in Fourier theory, so I looked for a function that might have this as its Fourier series.
Let , and define a function by
Make f into an odd function on the interval by defining Notice that when
Since f is odd, its Fourier cosine coefficients will all be 0. The sine coefficients will be given by
Grind out thtose integrals, using integration by parts as necessary. You will find that if n is odd, and if n is even. The function f is continuous and piecewise smooth, so it is the sum of its Fourier series. Therefore
Now put and to see that That's close enough to the answer that I needn't continue.