Could you please help me with the following (we're allowed to use the MCT and DCT):

Justifying your calculations, show that if 0<a<b then

\(\displaystyle \int_{-\infty}^{\infty} \frac{sinh(ax)}{sinh(bx)} dx = 4a ( \frac{1}{b^2-a^2} + \frac{1}{9b^2 - a^2} +\frac{1}{25b^2 - a^2} + ... )\)

Now, expanding the numerator as a Taylor series we get \(\displaystyle \frac{\sum_{n=0}^{\infty} x^{2n+1}}{(2n+1)! sinh (bx)} \), and as each \(\displaystyle f_n=\frac{x^{2n+1}}{(2n+1)! sinh (bx)}\) is non-negative, we will be able to apply the MCT... but how do you integrate the \(\displaystyle f_n \)'s? And where do you go from here? Any help is appreciated.