# Thread: Show 1/(exp(x) -a) sin x is integrable over [0, infinity)

1. ## Show 1/(exp(x) -a) sin x is integrable over [0, infinity)

I can show that for n > 0, $\displaystyle exp(-nx) \sin x$ is Lebesgue integrable over $\displaystyle [0,\infty)$ and that $\displaystyle \int_{0}^\infty exp(-nx) \sin dx = (1+n^2)^{-1}$.

Now I need to show that for $\displaystyle 0 \leq a \leq 1, (exp(x) -a )^{-1} sin x$ is integrable over $\displaystyle [0, \infty)$ and that

$\displaystyle \int_{0}^\infty (exp(x) -a)^{-1} \sin x dx = \sum a^{n-1}/(1+n^2)$, where the sum is from n=1 to infinity. I'm struggling over both the last bits - what's the idea here?

2. ## Taylor Series Substitution

$\displaystyle \int_0^\infty \frac1{e^x-a}\sin x dx$ = $\displaystyle \int_0^\infty e^{-x}\frac{1}{1-ae^{-x}}\sin x dx$ = $\displaystyle \int_0^\infty e^{-x}(\sum_{n=1}^\infty a^{n-1}e^{-(n-1)x})\sin x dx$ = $\displaystyle \sum_{n=1}^\infty a^{n-1}\int_0^\infty e^{-nx}\sin x dx$ = $\displaystyle \sum_{n=1}^\infty a^{n-1}\frac1{1+n^2}$ QED