On the approximation $\pi\approx 31^\frac{1}{3}$

Question

The Mathworld page about approximations to $\pi$ indicates that M. Joseph communicated the approximation

$$\pi \approx 31^\frac{1}{3}$$

in 2006, which is listed as formula 43 in http://mathworld.wolfram.com/PiApproximations.html.

Are there further references to this approximation or its author?

Answer 1

Such approximation is a consequence of a well-known identity: $$ \frac{\pi^3}{32} = \sum_{n\geq 0}\frac{(-1)^{n}}{(2n+1)^3} \tag{1}$$ The RHS of $(1)$ is a fast-convergent series and $32\sum_{n=0}^5 \frac{(-1)^{n}}{(2n+1)^3}\approx 31$, hence $\color{red}{\pi^3\approx 31}$.

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In a similar fashion $\frac{5\pi^5}{1536}=\sum_{n\geq 0}\frac{(-1)^n}{(2n+1)^5}$ and $1536\sum_{n=0}^{6}\frac{(-1)^n}{(2n+1)^5}\approx 1530+\frac{1}{10}$ lead to $$ \pi^5 \approx \frac{15301}{50} \tag{2} $$ but I guess that $\pi\approx\left(\frac{15301}{50}\right)^{\frac{1}{5}}$ is a less fascinating approximation than $\pi\approx 31^{1/3}$.

Answer 2

This is just the fact that $\pi^3\approx 31.00628$. I doubt you will find a reference.