Reciprocal times Derivative of Gamma Function/Examples/Sum of Reciprocal of 1 + 3k Alternating in Sign

Example of Use of Reciprocal times Derivative of Gamma Function

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

Proof

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

\(=\)

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

Reciprocal times Derivative of Gamma Function: Corollary $2$

\(\ds \leadsto \ \ \)

\(\ds 6 \sum_{k \mathop = 1}^\infty \dfrac {\paren {-1}^{k + 1} } {1 + 3 k}\)

\(=\)

\(\ds \map \psi {\dfrac 1 6 + 1} - \map \psi {\dfrac 1 6 + \dfrac 1 2}\)

$a := 1$ and $b := 3$

\(\ds \leadsto \ \ \)

\(\ds \sum_{k \mathop = 1}^\infty \dfrac {\paren {-1}^{k + 1} } {1 + 3 k}\)

\(=\)

\(\ds \dfrac {\map \psi {\dfrac 7 6} - \map \psi {\dfrac 2 3} } 6\)

dividing both sides by $6$

\(\ds \)

\(=\)

\(\ds \dfrac {\paren {-\gamma - 2 \ln 2 - \dfrac 3 2 \ln 3 - \dfrac {\pi \sqrt 3} 2 + 6} - \paren {-\gamma - \dfrac 3 2 \ln 3 + \dfrac \pi {2 \sqrt 3} } } 6\)

Digamma Function of Seven Sixths and Digamma Function of Two Thirds

\(\ds \)

\(=\)

\(\ds \paren{ -\dfrac \gamma 6 + \dfrac \gamma 6} - \dfrac 1 3 \ln 2 + \paren {-\dfrac 3 2 \ln 3 + \dfrac 3 2 \ln 3} + \paren {-\dfrac {\pi \sqrt 3} {12} - \dfrac {\pi \sqrt 3} {36} }\)

grouping terms

\(\ds \)

\(=\)

\(\ds 1 - \dfrac 1 3 \ln 2 - \dfrac {\pi \sqrt 3} 9\)

$\blacksquare$