Remark
This is a homework type of question and since there is no homework tag anymore I ask you to tell me how would you solve this (hints) and let me solve it for myself so I can practice.
I can post the solution afterwards so next generations would have access to both hints and the full solution.
Problem
Show uniform convergence of a series
$$ \sum_{k}\frac{1}{k} \sin \left ( \frac{\pi k^2}{x + k} \right ) $$ where $x \in \left ( 0, \frac{\pi}{2} \right )$. Use the identity $\sin(y - k \pi) = (-1)^k \sin(y)$.
Partial solution
I defined $$ f_k(x) = \frac{1}{k} \sin \left ( \frac{\pi k^2}{x + k} \right ) $$ and used the identity where $y = \frac{\pi k^2}{x + k}$.
That gives
$$ f_k(x) = \frac{(-1)^{k+1}}{k} \sin \left ( \frac{\pi k x}{x + k} \right ). $$
Now I want to apply Dirichlet's theorem where $g_k(x) = (-1)^{k+1}$ which has bounded partial sums as needed.
Problem is with $h_k(x) = \frac{\sin \left ( \frac{\pi k x}{x + k} \right )}{k}$. It has to converge (as sequence) to zero. No problem at all if you consider $|h_k(x)| \leq \frac{1}{k} \rightarrow 0$ for $k \rightarrow \infty$. The real problem is with proving monotonicity of $h_k(x)$ (as a sequence) for $\forall x \in \left ( 0, \frac{\pi}{2} \right )$.
Tried & failed
I've tried to calculate $\frac{\partial h_k(x)}{\partial k}$ and calculate it's stationary points.
$\frac{\partial h_k(x)}{\partial k}$ yields
$$ \frac{ \cos \left ( \frac{\pi k x}{x+k} \right ) \frac{\pi k x^2}{(k+x)^2} - \sin \left ( \frac{\pi k x}{x + k} \right )}{k^2} $$ so $\frac{\partial h_k(x)}{\partial k} = 0$ is a nonlinear equation I am not able to solve.
The equation looks like this (after a few rearrangements)
$$ \cos \left ( \frac{\pi k x}{x+k} \right ) \pi k x^2 = \sin \left ( \frac{\pi k x}{x+k} \right ) (k+x)^2 $$
Questions
- Is there a way to solve that equation?
- Is there another way to prove monotonicity of the function $h_k(x)$?
- Is there another way to prove convergence of the series $\sum_k f_k(x)$?