# Find the limit of $‎\mu_n(x)=‎x\psi(n+1)‎+\sum_{k=1}^n‎\left(‎\psi(k+1)-‎\psi(k+1+x)\right)$?

‎Consider the functional sequence ‎$$‎\mu_n(x)=‎x\psi(n+1)‎+\sum_{k=1}^n‎‎‎‎\left(‎\psi(k+1)-‎\psi(k+1+x)\right)$$ where ‎$‎‎\psi(x)$ ‎is‎ the digamma function. I know that ‎‎$‎\mu_n(x)‎$‎ is convergent on ‎$‎(-2,+\infty‎)‎$‎. ‎‎‎Anyone can help me to find its limit function?