ゼータ関数の交代級数

\begin{align*} \sum_{k=1}^{\infty}\left(\zeta\left(2k\right)-\zeta\left(2k+1\right)\right) & =\sum_{k=1}^{\infty}\sum_{j=1}^{\infty}\left(\frac{j}{j^{2k+1}}-\frac{1}{j^{2k+1}}\right)\\ & =\sum_{k=1}^{\infty}\sum_{j=2}^{\infty}\left(\frac{j-1}{j^{2k+1}}\right)\\ & =\sum_{j=2}^{\infty}\frac{j-1}{j^{3}}\sum_{k=1}^{\infty}\left(\frac{1}{j^{2}}\right)^{k-1}\\ & =\sum_{j=2}^{\infty}\frac{j-1}{j^{3}}\frac{1}{1-\frac{1}{j^{2}}}\\ & =\sum_{j=2}^{\infty}\frac{j-1}{j^{3}-j}\\ & =\sum_{j=2}^{\infty}\frac{j-1}{j(j+1)(j-1)}\\ & =\sum_{j=2}^{\infty}\frac{1}{j(j+1)}\\ & =\sum_{j=2}^{\infty}\left(\frac{1}{j}-\frac{1}{j+1}\right)\\ & =\frac{1}{2} \end{align*}