Is there a name for this series?
$$\sum_{k=1}^{\infty}\frac{a^{2k}}{2k}.$$
I know that:
$$\tanh^{-1}(a)=\sum_{k=1}^{\infty}\frac{a^{2k-1}}{2k-1}.$$
Is there a name for this series?
$$\sum_{k=1}^{\infty}\frac{a^{2k}}{2k}.$$
I know that:
$$\tanh^{-1}(a)=\sum_{k=1}^{\infty}\frac{a^{2k-1}}{2k-1}.$$
I don't believe there's a name for the series you have there, but,
$$\begin{align*} -\log(1-z)&=\sum_{k=1}^\infty\frac{z^k}{k}\\ -\log(1-z^2)&=\sum_{k=1}^\infty\frac{z^{2k}}{k}\\ -\frac12\log(1-z^2)&=\sum_{k=1}^\infty\frac{z^{2k}}{2k}\\ \log\frac1{\sqrt{1-z^2}}&=\sum_{k=1}^\infty\frac{z^{2k}}{2k}\\ \end{align*}$$