Digamma at n+1

Theorem

The following formula holds: $$\psi(n+1)=1+\dfrac{1}{2}+\dfrac{1}{3}+\ldots+\dfrac{1}{n} - \gamma=H_n - \gamma,$$ where $\psi$ denotes the digamma function and $\gamma$ denotes the Euler-Mascheroni constant, and $H_n$ is the $n$th harmonic number.

References

• 1953: Harry Bateman: Higher Transcendental Functions Volume I ... (previous) ... (next): $\S 1.7 (9)$
• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $6.3.2$