Reciprocal times Derivative of Gamma Function/Examples

Examples of Use of Reciprocal times Derivative of Gamma Function

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

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

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

$\ds \sum_{k \mathop = 1}^\infty \dfrac {\paren {-1}^{k + 1} } {1 + 2 k} = 1 - \dfrac \pi 4$

Example: $\ds \sum_{k \mathop = 1}^\infty \dfrac {\paren {-1}^{k + 1} } {1 + 3 k}$

$\ds \sum_{k \mathop = 1}^\infty \dfrac {\paren {-1}^{k + 1} } {1 + 3 k} = 1 - \dfrac 1 3 \ln 2 - \dfrac {\pi \sqrt 3} 9$

Example: $\map {H^{\paren 1} } {\dfrac 1 2}$

$\harm 1 {\dfrac 1 2} = 2 - 2 \ln 2$

Example: $\map {H^{\paren 1} } {-\dfrac 1 2}$

$\harm 1 {-\dfrac 1 2} = -2 \ln 2$