Sum of Fourier series members with odd indices

Question

I have the following Fourier series for the function $f(x) = e^x$ on $[-\pi, \pi]$ $$ \frac{e^{\pi} - e^{-\pi}}{2\pi} + \sum_{n = 1}^{\infty}\left[\frac{(-1)^n(e^{\pi} - e^{-\pi})}{\pi(n^2 + 1)}\cos nx + \frac{n(-1)^n(e^{-\pi} - e^{\pi})}{\pi(n^2 + 1)} \sin nx\right] $$ I want to find the following sum on $(-\pi, \pi)$: $$ \sum_{n = 0}^{\infty}(a_{2n + 1}\cos(2n + 1)x + b_{2n + 1}\sin(2n + 1)x) $$ How can I do it ?

Answer 1

Hint

Shift $f(t)$ as much as half its period to obtain $f(t-\pi)$ and then find the Fourier series of $f(t)-f(t-\pi)$.