纯数学教程 Page 325 例LXVIII (14) $\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\cdots=\frac{3}{4}(\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots)$

\begin{equation}\label{eq:fiick}
\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\cdots=\frac{3}{4}(\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots)
\end{equation}

证明:
\begin{align*}
\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdots&=(\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\cdots)+(\frac{1}{2^2}+\frac{1}{4^2}+\frac{1}{6^2}+\cdots)\\&=(\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\cdots)+\frac{1}{2^2}(\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots)
\end{align*}
因此\ref{eq:fiick}成立.

\begin{equation}\label{eq:killed}
\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{6^2}+\frac{1}{7^2}+\frac{1}{9^2}+\cdots=\frac{15}{16}(1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots)
\end{equation}

证明:
\begin{align*}
\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{6^2}+\frac{1}{7^2}+\frac{1}{9^2}+\cdots&=(\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots)-(\frac{1}{4^2}+\frac{1}{8^2}+\frac{1}{12^2}+\cdots)\\&=(\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots)-\frac{1}{4^2}(\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots)
\end{align*}

因此\ref{eq:killed}成立.