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Continued fractions of square roots

Let’s look at the continued fraction representation for √14.

sqrt{14} = 3 + cfrac{1}{1+ cfrac{1}{2+ cfrac{1}{1+ cfrac{1}{6+ cfrac{1}{1+ cfrac{1}{2+ cfrac{1}{1+ cfrac{1}{6+ ddots}}}}}}}}sqrt{14} = 3 + cfrac{1}{1+ cfrac{1}{2+ cfrac{1}{1+ cfrac{1}{6+ cfrac{1}{1+ cfrac{1}{2+ cfrac{1}{1+ cfrac{1}{6+ ddots}}}}}}}}

If we were to take more terms, the sequence of denominators would repeat:

1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, …

We could confirm this with Mathematica:

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Compact form of the Lagrange inversion formula

The Lagrange inversion formula can be used to find the power series for the inverse of a function. I wrote about a different approach this problem a couple years ago, that time using Bell polynomials. This time I’ll give a formula that is more direct and easier to remember.

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Accelerating an alternating series

This post looks at an algorithm by Cohen et al [1] to accelerate the convergence of an alternating series. This method is much more efficient than the classical Euler–Van Wijngaarden method.

For our example, we’ll look at the series

sum_{k=1}^infty frac{(-1)^k}{k^2}sum_{k=1}^infty frac{(-1)^k}{k^2}

which converges slowly to -π²/12.

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Concentric circle images go wild

HAKMEM Item 123 gives two plots, both images of concentric circles under functions given by power series with rapidly thinning terms [1]. The first is the function

f(z) = sum_{n=1}^infty frac{z^{n!}}{n!}f(z) = sum_{n=1}^infty frac{z^{n!}}{n!}

and the second is

 g(z) = sum_{n=1}^infty frac{z^{2^n}}{2^n} g(z) = sum_{n=1}^infty frac{z^{2^n}}{2^n}

The lower limits of summation are not given in the original.

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Is there a zip code that equals its population?

US stamp from 1973 promoting zip codesUS stamp from 1973 promoting zip codes

I noticed yesterday that the population in a zip code near me is roughly equal to the zip code itself. So I wondered:

Does any zip code equal its population?

Yes, it’s a silly question. A zip code isn’t a quantity.

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Symbol pronunciation in Unix culture

I was explaining to someone this evening that I’m in the habit of saying “bang” rather than “exclamation point.” Here’s a list of similar nicknames for symbols.

These nicknames could complement the NATO phonetic alphabet if you needed to read symbols out loud,

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A different kind of computational survival

Abandoned shopping mallAbandoned shopping mall

Last year I wrote a post about being a computational survivalist, someone able to get their work done with just basic command line tools when necessary. This post will be a different take on the same theme.

I just got a laptop from an extremely security-conscious client.

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Not quite going in circles

foggy pathfoggy path

Sometimes you feel like you’re running around in circles, not making any progress, when you’re on the verge of a breakthrough. An illustration of this comes from integration by parts.

A common example in calculus classes is to integrate ex sin(x) using integration by parts (IBP).

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The things in proofs are weird: a thought on student difficulties

By Ben Blum-Smith, Contributing Editor

“The difficulty… is to manage to think in a completely astonished and disconcerted way about things you thought you had always understood.” ― Pierre Bourdieu, Language and Symbolic Power, p. 207

Proof is the central epistemological method of pure mathematics,

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A Geometric Approach to Functions

by Karen Hollebrands, Allison McCulloch, Daniel Scher, and Scott Steketee

Fostering an understanding and appreciation of the deep, beautiful threads that unite seemingly disparate areas of mathematics is among the most valuable outcomes of teaching. Two such areas that are ripe for bridge building—functions and geometric transformations—are the focus of our NSF project,

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