A question on the approximation of $ln(x!)$ for small x
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In a question of Arfken and Weber's Mathematical Methods for Physicists, For small values of $x$ , $ln(x!)=-gamma x+sumlimits_{n=2}^{infty}(-1)^nfrac{zeta(n)}{n}x^n$ , where the symbols used have their usual meanings. Now I need to prove that this can also be written as $$ln(x!)=frac{1}{2}lnleft(frac{pi x}{sinpi x}right)-gamma x-sumlimits_{n=1}^{infty}frac{zeta(2n+1)}{2n+1}x^{2n+1}$$ I can only do this. $$sumlimits_{n=2}^{infty}(-1)^nfrac{zeta(n)}{n}x^n=\ left(frac{zeta(2)}{2}x^2+frac{zeta(4)}{4}x^4+frac{zeta(6)}{6}x^6+...right)-left(frac{zeta(3)}{3}x^3+frac{zeta(5)}{5}x^5+frac{zeta(7)}{7}x^7+...right) =\sumlimits_{n=1}^{infty}frac{zeta(2n)}{2n}x^{2n}-sumlimits_{n=1}^{infty}frac{zeta(2n+1)}{2n+1}x^{2n+1}$$ Putting this in the original equation and comparing it with what needs to be p...
