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\begin{align}
\lim_{p \to \infty}\sum_{n = 1}^{p}
\sum_{k = 0}^{\infty}{\pars{-1}^{n + k} \over n\pars{2k + 1}^{n}} & =
\lim_{p \to \infty}\sum_{n = 1}^{p}{\pars{-1}^{n} \over n}
{1 \over \ic}\sum_{k = 0}^{\infty}{\ic^{2k + 1} \over \pars{2k + 1}^{n}}
\\[5mm] &=
\lim_{p \to \infty}\sum_{n = 1}^{p}{\pars{-1}^{n} \over n}\,
{1 \over \ic}\sum_{k = 1}^{\infty}{\ic^{k} \over k^{n}}\,
{1 - \pars{-1}^{k} \over 2} =
\lim_{p \to \infty}\Im\sum_{n = 1}^{p}{\pars{-1}^{n} \over n}\,
\sum_{k = 1}^{\infty}{\ic^{k} \over k^{n}}
\\[5mm] &=
\lim_{p \to \infty}\Im\sum_{k = 1}^{\infty}\ic^{k}
\sum_{n = 1}^{p}{\pars{-1/k}^{n} \over n} =
-\Im\sum_{k = 1}^{\infty}\ic^{k}\ln\pars{1 + {1 \over k}}
\\[5mm] & =
-\,\Im\sum_{k = 1}^{\infty}\ic^{k}\int_{0}^{1}{\dd t \over t + k} =
-\int_{0}^{1}\Im\sum_{k = 1}^{\infty}{\ic^{k} \over k + t}\,\dd t
\\[5mm] & =
-\,{1 \over 2}\int_{0}^{1}
\sum_{k = 0}^{\infty}{\pars{-1}^{k} \over k + 1/2 + t/2}\,\dd t
\\[5mm] & =
{1 \over 4}\int_{0}^{1}
\sum_{k = 0}^{\infty}\pars{{1 \over k + 3/4 + t/4} -
{1 \over k + 1/4 + t/4}}\,\dd t
\\[5mm] & =
{1 \over 4}\int_{0}^{1}\bracks{\Psi\pars{{t \over 4} + {1 \over 4}} -
\Psi\pars{{t \over 4} + {3 \over 4}}}\,\dd t\quad
\pars{~\substack{\Psi:\ Digamma\ Function}~}
\\[5mm] & =
\left.\ln\pars{\Gamma\pars{t/4 + 1/4} \over \Gamma\pars{t/4 + 3/4}}
\right\vert_{\ 0}^{\ 1} =
\ln\pars{{\Gamma\pars{1/2} \over \Gamma\pars{1}}
\,{\Gamma\pars{3/4} \over \Gamma\pars{1/4}}}
\\[5mm] & =
\ln\pars{\root{\pi}
\,{\Gamma^{\,2}\pars{3/4} \over \Gamma\pars{3/4}\Gamma\pars{1/4}}} =
\ln\pars{\root{\pi}\Gamma^{\,2}\pars{3/4} \over \pi/\sin\pars{\pi/4}}
\\[5mm] & =
\bbx{\ln\pars{\Gamma^{\,2}\pars{3/4} \over \root{2\pi}}} \approx -0.5123
\end{align}