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An old problem from Whittaker and Watson I'm having issues with. Any guidance would be appreciated.

Show that the function $$ f(x)=\int_0^\infty \left\{ \log u +\log\left(\frac{1}{1-e^{-u}} \right) \right\}\frac{du}{u}e^{-xu} $$ has the asymptotic expansion $$ f(x)=\frac{1}{2x}-\frac{B_1}{2^2x^2}+\frac{B_3}{4^2x^4}-\frac{B_5}{6^2x^6}+\;... \;, $$ where $$B_1, B_3, ...$$ are Bernoulli's numbers.

Show also that f(x) can be developed as an absolutely convergent series of the form $$ f(x)=\sum_{k=1}^\infty\frac{c_k}{(x+1)(x+2)...(x+k)} $$

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