Help with two functions - continuity, Laplace transform and Fourier series

Question

I've been practicing for my exam lately, and there are two function that I've had a real trouble analyzing.

1.
$f(x) = \sum_{n=1}^{\infty} \frac{\sin(nx)}{10^n \sin(x)}$, for $x \neq k\pi$
$f(x) = \lim_{x\to k\pi} f(x)$, for $x = k\pi$

a) check continuity of f(x) and its derivative
b) find its Fourier series

2. $g(x) = \int_0^\infty \frac{\cos(tx)}{t^2 + a^2}dt$, $(x \ge 0, a \gt 0)$

a) check continuity of g(x)
b) find its Laplace transform.
c) apply the inverse Laplace transform to the function you got in b)

Answer 1

1.) If we set $f_n(x)=\frac{\sin(nx)}{\sin x}$ and define $f_n$ over $\pi\mathbb{Z}$ as $n$, $f_n(x)=U_{n-1}(\cos x)$ is a $2\pi$-periodic and $C^{\infty}$ function for which: $$ \left|f_n(x)\right|\leq n,\qquad \left|f_n'(x)\right|\leq \frac{4}{\pi^2}(n+1)^2.$$ Since both $\frac{n}{10^n}$ and $\frac{n^2}{10^n}$ are summable over $n\geq 1$, it follows that both: $$ f(x) = \sum_{n\geq 1}\frac{f_n(x)}{10^n},\qquad f'(x)=\sum_{n\geq 1}\frac{f_n'(x)}{10^n} $$ are continous and bounded functions over $\mathbb{R}$. Obviously: $$ \lim_{x\to k\pi}f(x)=\sum_{n\geq 1}\frac{n}{10^n}=\frac{10}{81}.$$ In order to compute the Fourier series of $f(x)$, it is useful to notice that $f(x)$ is an even function and that: $$ \int_{-\pi}^{\pi}\frac{\sin(nx)}{x}\cos(mx)\,dx = 2\pi $$ if $m$ is a non-negative integer less than $n$ and with opposite parity (otherwise, the previous integral is just zero). That gives:

$$ f(x)=-\frac{10}{99}+\sum_{m\geq 0}2\cos(mx)\sum_{k\geq 0}\frac{1}{10^{m+2k+1}}=\frac{10}{99}+\frac{20}{99}\sum_{m\geq 1}\frac{\cos(mx)}{10^{m}}. $$

2.) This is a classical problem. By the residue theorem: $$ g(x)=\int_{0}^{+\infty}\frac{\cos(tx)}{t^2+a^2}\,dt = \frac{\pi}{2a}\,e^{-ax}$$ and the other points are trivial.