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$$\int_{0}^{1}log(\Gamma (x))dx$$

and the last step To solve the problem is:

$$1 -\frac{\gamma }{2} + \lim_{n\rightarrow \infty } \frac{H_{n}}{2} + n + log(\Gamma (n+1)) - (n+1)(log(n+1))$$

and wolfram alpha tells me something about series expansion at ##n=\infty## of laurent series

http://www.wolframalpha.com/input/?i=limit&rawformassumption={"F",+"Limit",+"limitfunction"}+->"H_n/2+++n+++ln(n!)+-+(n+1)ln(n+1)"&rawformassumption={"F",+"Limit",+"limitvariable"}+->"n"&rawformassumption={"F",+"Limit",+"limit"}+->"infinity"&rawformassumption={"FVarOpt"}+->+{{"Limit",+"direction"},+{"Limit",+"limitvariable"}}&rawformassumption={"C",+"limit"}+->+{"Calculator"}

I know a little about series of laurent, but I do not understand how they serve to solve limits

and expansion at ##n=\infty##.