I am looking at a certain derivation for the full Fourier series of $\phi(x)=\cosh{x}$ on the interval $(-l,l)$. The derivation uses the definition of hyperbolic cosine and the full Fourier series for $e^x$ as follows
$$\begin{align} \cosh{x}&=\frac{e^x+e^{-x}}{2} \\ &=\frac{1}{2}\left(\sum_{n=-\infty}^\infty (-1)^n \frac{l+in\pi}{l^2+n^2\pi^2}e^{\frac{in\pi x}{l}} \sinh{l}+\sum_{n=-\infty}^\infty (-1)^n \frac{l+in\pi}{l^2+n^2\pi^2}e^{-\frac{in\pi x}{l}} \sinh{l}\right)\\&= \frac{1}{2}\left(\sum_{n=-\infty}^\infty (-1)^n \frac{l+in\pi}{l^2+n^2\pi^2}e^{\frac{in\pi x}{l}} \sinh{l}+\sum_{n=-\infty}^\infty (-1)^{-n} \frac{l-in\pi}{l^2+n^2\pi^2}e^{\frac{in\pi x}{l}} \sinh{l}\right) \end{align}$$
My confusion is in seeing how the second sum on the middle line is apparently equal to the second sum on the last line. I have thought very long about this and may be missing something simple, but would really appreciate any help in seeing why this is true. Thanks in advance.