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We wish to show that for small $x$, $$\sqrt{1+x} \approx 1 + \frac{x}{2}$$ To derive this formula, we use the well known fact that $$(1+x)^\alpha = \sum_{n=0}^\infty \frac{x^n}{n!} \prod_{k=0}^{n-1} (\alpha - k)$$ The result follows by substituting $\alpha = 1/2$ and truncating the series after two terms.

I gave a talk a USF a while back for the Special Lecture Series in Computer Science. The goal was to try to give a flavor for the intuition behind simulation algorithms. I’m not sure I succeeded in doing this coherently, but good practice for the next time.

Here are the slides, in pdf and keynote form. Caution: they are not very self-contained.

A few days ago I was in another discussion where someone raised the question of why presidential elections are so close. In the interests of avoiding these discussions in future, here’s a histogram of the popular vote margin over all presidential elections (data from [1]):

Conclusion: presidential elections aren’t very close, and we should stop looking for explanations of why they are.