Simplifying big expression

Question

What to do with this?

$$f(x) = \frac{\sinh(\pi)}{\pi} + \frac{2\sinh(\pi)}{\pi}\sum_{n=1}^\infty (-1)^n \left[\frac{\cos(nx)-n \sin(nx)}{1 + n^2}\right]$$

Can it be simplified?

Answer 1

We have: $$\int_{-\pi}^{\pi}e^x \cos(n x)\,dx = 2\sinh(\pi)\frac{(-1)^n}{1+n^2}$$ $$\int_{-\pi}^{\pi}e^x \sin(nx)\,dx = 2\sinh(\pi)\frac{(-1)^{n+1} n}{1+n^2}$$ hence over $I=(-\pi,\pi)$ we have: $$ e^x = \frac{\sinh \pi}{\pi}+2\frac{\sinh \pi}{\pi}\sum_{n=1}^{+\infty}(-1)^n\frac{\cos(nx)-n\sin(nx)}{1+n^2}$$ and the given series is just the Fourier series of $e^x$ over $I$.