Euler's Approximation of pi.

Question

I recently stumbled across the formula:

$$\pi=20\arctan\frac{1}{7}+8\arctan\frac{3}{79}$$

developed by Euler, for approximating pi. I evaluated it to several thousand decimal places and up to that point, it accurately represented pi.

When does this equation break down in its representation of pi?

2 related questions: http://math.stackexchange.com/questions/297/simple-numerical-methods-for-calculating-the-digits-of-pi and http://math.stackexchange.com/questions/14113/series-that-converge-to-pi-quickly — Jonas Meyer, Jan 11 '12 at 04:20
A derivation of the identity and related ones can be found here — t.b., Jan 11 '12 at 04:43
Quick proof: The complex number $(7 + i)^{20} (79 + 3 i)^8 = -149011611938476562500000000000000$ lies on the negative real axis, so $20\arctan\frac{1}{7}+8\arctan\frac{3}{79} = \pi + 2\pi n$. Consideration of the sizes of the angles involved shows that $n$ must be zero. — Hans Lundmark, Jan 11 '12 at 07:46
$-149011611938476562500000000000000 = 2^{14}5^{40}$, just for info. — Tito Piezas III, Oct 05 '13 at 02:55

Jonas Meyer · Accepted Answer · 2012-01-12T07:49:16.947

That formula doesn't break down. It can be proven that the right-hand side is equal to $\pi$. One reason it comes up in connection with approximation is that the arctangent terms can be approximated well by adding terms from the Taylor expansion of the arctangent function,

$$\arctan(x)=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\frac{x^9}{9}-\cdots.$$

This is valid for $|x|\leq 1$, but the series converges more rapidly the closer $x$ is to $0$, so with the relatively small $\frac{1}{7}$ and $\frac{3}{79}$ as inputs it gives a good approximation without having to take too many terms. This can be contrasted with the formula $$\pi=4\arctan(1)=4\left(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots\right),$$ where the series converges much more slowly.

In 1950 H.C. Schepler published a 3 part "Chronology of pi" in Mathematics Magazine, available to those with access to JSTOR here, here, and here. In the second part there is the following excerpt indicating that Hutton may have suggested using the formula before Euler:

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Schepler's chronology can also be found in L. Berggren, J. Borwein, and P. Borwein's Pi: A Source Book.

Thanks for explaining why it's associated with an 'approximation'. That had me a bit confused — Chris A, Jan 11 '12 at 12:44

score 1 · Answer 2 · answered Jan 11 '12 at 11:18

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The Soviet 8th grade students usually solve this problem as follows. First, divide by 4. Then a portable arc tangent in the opposite direction. Then calculate the tangent of the right and left part using very simple formula tg(A+B).

answered Jan 11 '12 at 11:18

Dmitri

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"A portable arc tangent" ... not a correct translation into English of what you want to say. – GEdgar Jan 11 '12 at 14:05
What is a portable arc tangent? – Rudy the Reindeer Apr 07 '13 at 19:52

Euler's Approximation of pi.

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