Uniform convergence of $\sum_{n=1}^{\infty} \frac{\sin(n x) \sin(n^2 x)}{n+x^2}$

Solution 1:

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.