잡담을 위한 소박한 장소

Problem. Prove the following identities. \begin{equation}\label{eqn:wts} \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^3 \binom{2n}{n}} = \frac{2}{5} \zeta(3), \qquad \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^3 2^n \binom{2n}{n}} = \frac{1}{4} \zeta(3) - \frac{1}{6}\log^3 2 \end{equation} We divide the proof into several lemmas. Lemma 1. For $|x| < 1$. \begin{equation}\label{eqn:wts_lem_1} \int_{0}^{x} ..