<s>[INST] Uniform convergence of $\sum_{n=1}^{\infty} \frac{\sin(n x) \sin(n^2 x)}{n+x^2}$ I'm not sure wether or not the following sum uniformly converge on $\mathbb{R}$ :
$$\sum_{n=1}^{\infty} \frac{\sin(n x) \sin(n^2 x)}{n+x^2}$$
Can someone help me with it? (I can't use Dirichlet' because of the areas where $x$ is close to $0$)
 [/INST] The series does converge uniformly. For the proof, put $S_n(x) = \sum_{k = 0}^n \sin{(kx)}\sin{(k^2 x)}$ for $n\geq 0$. The general idea is to use summation by parts to reduce ourselves to showing that $S_n(x)$ is bounded uniformly, and then to prove that by giving a closed form for $S_n(x)$.
First, the summation by parts (I write $S_n$ in place of $S_n(x)$ for brevity):
$$
\begin{align}
\sum_{n = 1}^N{\sin{(nx)}\sin{(n^2 x)}\over n + x^2} & = \sum_{n = 1}^N {1\over n+x^2}(S_n - S_{n-1}) \\
& = \sum_{n = 1}^N {S_n\over n+x^2} - \sum_{n = 1}^N {S_{n-1}\over n+x^2} \\
& = \sum_{n = 1}^N {S_n\over n+x^2} - \sum_{n = 0}^{N-1} {S_{n}\over n+1+x^2} \\
& = {S_N\over N+x^2} - {S_0\over 1 + x^2} + \sum_{n = 1}^{N-1} S_n\left({1\over n+x^2} - {1\over n+1+x^2}\right) \\
& = {S_N\over N+x^2} + \sum_{n = 1}^{N-1} {S_n\over (n+x^2)(n+1+x^2)}.
\end{align}
$$
From here it is clear that it is enough to prove that $S_n = S_n(x)$ is uniformly bounded in $x$.
To do this, note that 
$$
\begin{align}
2\sin{(kx)}\sin{(k^2 x)} &= \cos{\{(k^2 - k)x\}} - \cos{\{(k^2 + k)x\}} \\ 
& = \cos{\{k(k - 1)x\}} - \cos{\{(k+1)k)x\}},
\end{align}
$$ 
and therefore that
$$
\begin{align}
2S_n(x) & = \sum_{k = 0}^n 2\sin{(kx)}\sin{(k^2 x)} \\
& = \sum_{k = 0}^n\left(\cos{\{k(k - 1)x\}} - \cos{\{(k+1)k)x\}}\right)
\end{align}
$$
The last sum telescopes, and we are left with
$$
2S_n(x) = 1 - \cos{\{n(n+1)x\}},
$$
which is plainly bounded uniformly in $x$. So we're done.
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