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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?

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    $\begingroup$ I'm sure the values $\pi^2 \approx 9.87, \pi^3 \approx 31.006, \pi^4 \approx 97.409$ have been observed prior to 2006. Considering Ramanujan gave approximations for $\pi^4$, he must have known of the near-integer $\pi^3$ as well. $\endgroup$ – Tito Piezas III Jun 12 '17 at 14:03
  • $\begingroup$ math.stackexchange.com/questions/850442/… $\endgroup$ – Jack D'Aurizio Jun 12 '17 at 16:23
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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 $\color{red}{\pi^3\approx 31}$.


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 $\pi\approx 31^{1/3}$.

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  • $\begingroup$ With a fifth root I like $$\pi\approx 3+\frac{1}{17571^\frac{1}{5}}$$ Raymond Manzoni wrote it in a comment somewhere. $\endgroup$ – Jaume Oliver Lafont Jun 13 '17 at 12:41
  • $\begingroup$ We can write an exact expression for the error as: $$\pi^3 = 31 + \sum_{n=1}^\infty 5\frac{(-1)^n}{(2 n + 1)^3} + 27\frac{(-1)^{n + 1}}{(2 n + 3)^3}$$ $\endgroup$ – Jaume Oliver Lafont Jun 13 '17 at 14:39
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This is just the fact that $\pi^3\approx 31.00628$. I doubt you will find a reference.

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