This is a question from an old exam, I'm trying to understand the key and comments provided by the examiner.
First question is simply determine the Fourier series of the function $f$ defined as $1-x^2$ on $[-\pi,0)$ and as $1+x^2$ on $[0,\pi)$. I got $$f \sim 1+\frac{2}{\pi} \sum \frac{1}{n^3} \left((2-n^2 \pi^2)(-1)^n-2 \right) \sin{nx}$$
while the key gives $$f \sim 1+\frac{2}{\pi} \sum \frac{1}{n^3} \left((n^2 \pi^2-2)(-1)^{n+1}-2 \right) \sin{nx}$$
The only difference is style, but is it of any significance?
The key also has the following to say:
Note that $f(x) = 1 + f_1(x)$, where $f_1$ is odd.
This is the only mention of $f_1$, and I can't figure out what it could possibly refer to. Neither piecewise functions equals $1$ plus the other, so that can't be it. There has to be some significance to this comment but since I don't understand the comment in the first place said significance is lost on me.
Finally one is asked to sketch the Fourier series. Specifically:
Set $g(x) :=$ (the Fourier series of $f$) for $x$ in $\mathbb{R}$. Sketch the graph of $g$ on the interval $[-2\pi,2\pi]$. Be particularly clear at the jump points!
Well, I am dumbfounded. How does one go about finding even, say, $g(1)$, when you have to calculate an infinite number of values and sum them together?
Finally, what are the jump joints? Why isn't $g$ continuous everywhere despite $\sin{nx}$ being so?
Thank you.