Continued Fraction Expansion of Fourth Power of Pi

Theorem

The continued fraction expansion of the $4$th power of $\pi$ begins:

$\pi^4 \approx 97 + \cfrac 1 {2 + \cfrac 1 {2 + \cfrac 1 {3 + \cfrac 1 {1 + \cfrac 1 {16 \, 539} } } } }$

This sequence is A058286 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

The continued fraction expansion shown above evaluates to $\dfrac {2143} {22}$ or $\paren {97 \cdotp 4 \dot 0 \dot 9}$ which is $\approx 1.2 \times 10^{-7}$ lower than the exact value.

Proof

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Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $97 \cdotp 40909 \, 10340 \, 0 \ldots$