Reciprocal times Derivative of Gamma Function/Examples/Sum of Reciprocal of 1 + k Alternating in Sign

Example of Use of Reciprocal times Derivative of Gamma Function

$\ds \sum_{k \mathop = 1}^\infty \dfrac {\paren {-1}^{k + 1} } {1 + k} = 1 - \ln 2$

$\ds 2 b \sum_{k \mathop = 1}^\infty \dfrac {\paren {-1}^{k + 1} } {a + b k}$

$=$

$\ds \map \psi {\dfrac a {2 b} + 1} - \map \psi {\dfrac a {2 b} + \dfrac 1 2 }$

$\ds \leadsto \ \ $

$\ds 2 \sum_{k \mathop = 1}^\infty \dfrac {\paren {-1}^{k + 1} } {1 + k}$

$=$

$\ds \map \psi {\dfrac 1 2 + 1} - \map \psi {\dfrac 1 2 + \dfrac 1 2}$

$a := 1$ and $b := 1$

$\ds \leadsto \ \ $

$\ds \sum_{k \mathop = 1}^\infty \dfrac {\paren {-1}^{k + 1} } {1 + k}$

$=$

$\ds \dfrac {\map \psi {\dfrac 3 2 } - \map \psi 1 } 2$

dividing both sides by $2$

$\ds $

$=$

$\ds \dfrac {\paren {-\gamma - 2 \ln 2 + 2} - \paren {-\gamma} } 2$

$\ds $

$=$

$\ds 1 - \ln 2$

grouping terms

$\blacksquare$