Question

Examine whether the series

Answer 1

@SithsAndGiggles

Answer 2

@ganeshie8

Answer 3

@Alchemista

Answer 4

For a function \(f(x)\) with period \(L\), \(f(x)\) has a Fourier expansion if you can write it as
\[\frac{1}{2}a_0+\sum_{n=1}^\infty\left(a_n\cos\frac{n\pi x}{L}+b_n\sin\frac{n\pi x}{L}\right)\]
where
\[\begin{cases}
a_0=\displaystyle\frac{1}{L}\int_{-L}^Lf(x)\,dx\\\\
a_n=\displaystyle\frac{1}{L}\int_{-L}^Lf(x)\cos\frac{n\pi x}{L}\,dx\\\\
b_n=\displaystyle\frac{1}{L}\int_{-L}^Lf(x)\sin\frac{n\pi x}{L}\,dx
\end{cases}\]
For this particular summation, we want to show that \(a_0=0\) and \(a_n=b_n=1\) (where \(L=\pi\))...

Answer 5

From what I can tell, at the very least we know that the average value of whatever this \(f\) may be must be \(0\). If \(f\) were odd, we would of course get \(a_0=0\), but we'd also have that \(a_n=0\), since odd functions have Fourier series with only sine terms.

Answer 6

HI.. @SithsAndGiggles SO what is the conclusion..
I mean, how to examine.?

Answer 7

A function has some conditions to satisfy in order for it to have a Fourier expansion: http://en.wikipedia.org/wiki/Dirichlet_conditions

Plotting the first few terms of the series seems to suggest that as \(n\to\infty\), so does the number of extrema...