Exercise 2, § 2

Functions of a Real Variable, VII, 328

For every integer $k > 1$ put $S_k = \sum_{n=1}^\infty n^{-k}$ . Show that, for $-1 < x \leq 1$ one has

\log \Gamma(1 + x) = -\gamma x + \sum_{k=2}^\infty (-1)^k \frac{S_k}{k} x^k

the series on the right being uniformly convergent on every compact subset of ]

-1, 1

], and absolutely convergent for

|x| < 1

Solution

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