I just wanted to check my answer for 2 practice problems that I am doing, which follows from one another, the questions are as follows:
a) Find the coefficients in $c_n$ in the following fourier series: $$e^{x} = \sum_{n=-\infty}^{\infty}c_ne_n(x)$$ for $0 \leq x \leq 1$, and $n \in \mathbb{Z}$.
b) Use Parseval's formula to evaluate: $$\sum_{n=1}^{\infty}(1+4\pi^2n^2)^{-1}$$
So what I did is the following for (a): $$c_n = \int_{0}^{1}e^{-2\pi i n x}e^x dx$$ $$ = \int_0^1 e^{x(1-2\pi i n)}dx = \frac{1}{(1-2\pi i n)}e^{x(1-2\pi i n)}\bigg\vert_0^1$$ $$ = \frac{1}{(1-2\pi i n)}(e^{1-2\pi i n} - 1)$$ $$ = \frac{1}{(1-2\pi i n)}(e^{1}(1) - 1)$$ $$\vdots$$ $$c_n = \begin{cases} \frac{e^1 - 1}{(1-2\pi i n)} & \forall n, n\neq 0 \\ e - 1 & \forall n = 0 \end{cases}$$
Hence the fourier series is written as $$e^x = \sum_{n = -\infty}^{\infty}(\frac{e^1 - 1}{(1-2\pi i n)})e^{2\pi i n x}$$
For (b): It follows that $$\vert | f \vert |^2 = \int_{0}^{1}e^{2x}dx = \frac{1}{2}[e^2 - 1 ]$$ $$ \vdots $$ $$ \implies \frac{(e+1)}{4(e-1)} = \sum_{n=1}^{\infty}\frac{1}{(1+4\pi^2n^2)}$$
Does this look correct ?
Some check is greatly appreciated since I am trying to practice but dont have the solutions.
Thank you