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?
2026-03-27 00:05:57.1774569957
Find the limit of $\mu_n(x)=x\psi(n+1)+\sum_{k=1}^n\left(\psi(k+1)-\psi(k+1+x)\right)$?
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