# Strategies for proving continuity and differentiability of trigonometric series

Let $$f$$ be a function defined by a series $$f\left(x\right)=\sum c_n e^{inx}.$$ Sometimes, I can prove that the series converges pointwise (when it does), using the Dirichlet test. When the convergence is uniform by the Weierstrass test (i.e., $$\sum\left|c_n\right|<\infty$$), I can show that $$f$$ is continuous. When the series of derivatives converges uniformly ($$\sum\left|nc_n\right|<\infty$$), I can show that $$f$$ is differentiable. However, very often I plot the series, and $$f$$ looks clearly continuous and/or $$C^1$$ (sometimes piecewise, sometimes globally), but I can't prove it. What are some other tools, besides these basic theorems/tests I mentioned above?

Context: I want to show that $$f\left(x\right) = \sum_{n\geqslant 2} \frac{1-\cos nx}{n^2 \log n}$$ is $$C^1$$. It's enough to define $$g\left(x\right) = \sum_{n\geqslant 2} \frac{\sin nx}{n \log n},$$ show that this series converges (easy: Dirichlet test), $$g$$ is continuous (hard), and then integrate it (easy again). When I plot the graph, this series looks like it's uniformly convergent, but I have no idea how to show this. I would by satisfied with a proof that it is continuous at $$x=0$$, and that it conveges uniformly on any compact set away from zero. I can't show either of these things. My only clue is: I know that $$\sum \left(\sin nx\right)/n$$ is the sawtooth, which is discontinuous at zero, but with limits at the right and at the left. Multiplying the Fourier coefficients by $$1/\log n$$ looks like "smoothing", like a convolution, so it should be continuous indeed. How do I show this?

Let $$\{a_n\}_{n=1}^\infty \subset(0,\infty)$$ be non-increasing. Then the series $$\sum_{n=1}^\infty a_n \sin(nx)$$ is uniformly convergent if and only if $$\lim_{n \to \infty} n a_n =0$$.
This result shows that indeed your series defining $$g$$ converges uniformly, and so the rest of your argument goes through.