Pi Approximation Day

[Thomas Klemm]

For \(p=5\) we get:

\(
\begin{align}
\sum_{n=1}^{\infty} \frac{1}{(n (n + 1))^5} = 126 - \frac{35 \pi^2}{3} - \frac{\pi^4}{9}
\end{align}
\)

Depending on the number of terms used on the left-hand side, we get the following approximations for \(\pi\).

\(0\):

\(
\begin{align}
\sqrt{\frac{3}{2} \left(\sqrt{1729} - 35 \right)} \approx 3.14195300074
\end{align}
\)

\(\frac{1}{2^5}\):

\(
\begin{align}
\sqrt{\frac{3}{8} \left(\sqrt{27662} - 140 \right)} \approx 3.14159418065
\end{align}
\)

\(\frac{1}{2^5}+\frac{1}{6^5}\):

\(
\begin{align}
\frac{1}{6} \sqrt{\sqrt{5041398} - 1890} \approx 3.14159270392
\end{align}
\)