Posted by: Dave Richeson | December 9, 2008

A new continued fraction for pi

I love continued fractions.

The golden ratio:
\displaystyle\phi=1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\vdots}}}}}

The square root of 2:
\displaystyle \sqrt{2}=1+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{\vdots}}}}}

The base of the natural logarithm:
\displaystyle e=2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{4+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{6+\cfrac{1}{\vdots}}}}}}}}}

Pi:
\displaystyle \pi=3+\cfrac{1}{6+\cfrac{9}{6+\cfrac{25}{6+\cfrac{49}{6+\cfrac{81}{\vdots}}}}}

In the most recent American Mathematical Monthly (December 2008) Thomas J. Pickett and Ann Coleman, in their note “Another Continued Fraction for \pi,” present the following beautiful continued fraction in which the terms down the diagonal are the harmonic series.
\displaystyle\frac{\pi}{2}=1+\cfrac{1}{1+\cfrac{1}{1/2+\cfrac{1}{1/3+\cfrac{1}{1/4+\cfrac{1}{\vdots}}}}}

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Responses

  1. I’ve always been leery of continued fractions – I always have to think about how they’re evaluated. I figure out it’s bottom to top, but then I think the worst part is having to go through the sequence the whole way each time with each new element to see the result and how closely it’s approximating the final result. Unlike summation (and perhaps even product) series, I have no idea how these are developed or discovered. The book “Journey Through Genius” shows how Newton discovered several fast-converging summation series for Pi, and I found it fascinating reading.

    There is of course a well-known special notation for infinite summation using the Greek letter Sigma, and also one for infinite products using the capital Greek letter Pi, but not for other continued operations, such as radicals and division:

    http://en.wikipedia.org/wiki/Infinite_expression_%28mathematics%29

    If there were notations for these others, they might be more easily studied and understood, and more discoveries made about them.

    • As for notation b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) is the Gauss Notation for generalized continued fractions.

    • bradley, google up Gosper and continued fractions. There are term-at-a-time ways of evaluating them, which also yield term-at-a-time error bounds. Roughly, a linear fractional function consumes terms to create rationals; rationals can emit terms at a time (though this only completes the map cont.frac->rat->cont.frac). There is also a matrix interpretation, which is (not coincidentally) identical to the linear fractional function.

      Though it seems hard to believe looking at the form, the terms are indeed taken from left to right. Consider that a general continued fraction is (a + b/(c + r)) where “a”, “b”, and “c” are general terms and “r” is “the rest of the continued fraction.” Each approximation then ignores this remainder—a truncation.


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