I can't say much on this particular solution (I haven't had analysis yet), but I can say that another method I've seen to calculate this series (and which generalizes well to larger even n, I believe) is a combination of Fourier analysis/linear algebra/Parseval's identity (the latter is essentially the Pythagorean theorem for an arbitrary inner product space), as well as being quite accessible to undergrads with little analysis.
The following were questions on a lin. alg. quiz I took in high school:
Quote:
Originally Posted by Dr. Fogel
2) Compute the Fourier coefficients (for the interval [ , ]) of sines and cosines for the function f(x) = x.
....
4) Use Parseval's identity on the result from problem 2 to obtain an interesting result.

The answer to 2) is easily calculated to be 0 cosine coeffiecients, and the sine coefficients are
.
For problem 4), we apply Parseval's identity with our orthonormal basis
being
. We also note that
(where
), or rewriting to put it in the form of Parseval's identity,
. Thus
. Meanwhile,
, so we have
, or
.
The only squishy part (AFAICT) is proving Parseval's identity (esp. in the infinite dimensional case). We can also see why we get only even powers  because of the squaring of the fourier coefficients. It ought to be easy (though I haven't really thought about it) to use f(x)=x^2, x^3, ... to get
. (The only part that I haven't confirmed is that the Fourier coefficients take the form I think they take for the higher order f's.)