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**gammafunction** Hi! I would like to proof that $\displaystyle a_n:=\ln(n!)+n-(n+\frac{1}{2}) \ln(n)$ converges. I already have that

$\displaystyle (n+\frac{1}{2}) \ln(n+\frac{1}{2})-n \leq \ln(n!) \leq (n+1) \ln(n+1)-n$ or equivalently

$\displaystyle (n+\frac{1}{2}) \ln(1+\frac{1}{2n}) \leq \ln(n!)+n-(n+\frac{1}{2}) \ln(n) \leq (n+\frac{1}{2}) \ln(1+\frac{1}{n})+ \frac{1}{2} \ln(n+1) $

I have also calculated the difference $\displaystyle a_n - a_{n+1}=(n+\frac{1}{2}) \ln(1+\frac{1}{n})-1 $ but i do not know how to use my inequations.

Could somebody please help me with this?