Question-126732

Question Number 126732 by 0731619177 last updated on 23/Dec/20

Answered by Dwaipayan Shikari last updated on 24/Dec/20

\lim_{x \to 0} [ψ (x) + \frac{1}{x}] = - γ + \underset{n = 1}{\sum^{\infty}} \frac{1}{n} - \frac{1}{(n - 1) + x} + \frac{1}{x}

\underset{n = 1}{\sum^{\infty}} \frac{1}{n} - \frac{1}{(n - 1) + x} \sim - \frac{1}{x} (x i s v e r y s m a l l w i t h r e s p e c t t o \sum^{\infty} \frac{1}{n})

\underset{x \to 0}{\lim 1} - \frac{1}{x} + \frac{1}{2} - \frac{1}{1 + x} + \frac{1}{3} - \frac{1}{2 + x} + \dots . = - \frac{1}{x}

= - γ - \frac{1}{x} + \frac{1}{x} = - γ