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

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

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

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

    \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?
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  2. #2
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    Atlanta, GA
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    Taylor Series Substitution

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