Let $f(x)=\sum^\infty_{n=1}\frac{1}{2n^2-\sin(nx)}, x\in\mathbb R$
How do we decide whether $f$ is continuous on $\mathbb R$? And further, how can we tell if $f$ is differentiable?
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So far I have come to the conclusion that since $\frac{1}{2n^2-\sin(nx)}\le\frac{1}{2n^2-1}$, we can show that the series converges uniformly. We can then do a similar thing by considering $f(x)=\sum^\infty_{n=1}\frac{d}{dx}(\frac{1}{2n^2-\sin(nx)})$ and showing this also converges uniformly. How then can this be extended to show that $f$ is continuous on $\mathbb R$?
I believe that this can then be applied to show that the function is differentiable, but I am unsure on the details as my lecture notes for this part of the course are extremely sparse!
If anyone could help me to try to crack this I would much appreciate it!
The sum of a uniformly converging series of continuous functions is continuous.
If the terms have continuous derivatives and the series of derivatives converges uniformly, then that converges to a continuous function which is the derivative of the sum of the original series, so the sum of the original series is $C^1$, i.e. differentiable with continuous derivative.