Does the continued fractions $3+\frac{1}{5+\frac{1}{7+\cdots}}$ equal $\pi$?

Question

$3+\frac{1}{5+\frac{1}{7+\cdots}}=\pi$ Is it true? If yes, how to show it?

Please help. Thank you.

Answer 1

If the continued fraction converged to $\pi$, the value of the odd convergent $$ 3 +\cfrac1{ 5 +\cfrac1{ 7}} = \frac{115}{36} = 3.19444\ldots $$

would be strictly smaller than $\pi$. But the convergent is already too big. The actual value of the continued fraction must exceed $\frac{115}{36}$; it lies somewhere between $\frac{115}{36} \approx 3.1944$ and $\frac{1051}{329} \approx 3.1945$.

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Another way to see this is:

1. Simple continued fraction representations of irrational numbers are unique
2. The continued fraction representation of $\pi$ begins $3+\frac1{7+\cdots}$, not $3+\frac1{5+\cdots}$.
3. Therefore, this is not the continued fraction representation of $\pi$.