My colleague Tom Liggett recently posed to me the following problem about power series in one real variable. Observe that the power series has very rapidly decaying coefficients (of order ), leading to an infinite radius of convergence; also, as the series converges to, the series decays very rapidly as approaches. The […]

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Academics / Mathematics : The Unapologetic Mathematician

So, what’s so great right now about uniform convergence? As we’ve said before, when we evaluate a power series we get a regular series at each point, which may or may not converge. If we restrict to those points where it converges, ...

Academics / Mathematics : Statistical Modeling, Causal Inference, and Social Science

A colleague pointed me to a recent paper, “Does Regression Produce Representative Estimates of Causal Effects?” by Peter Aronow and Cyrus Samii, which begins: With an unrepresentative sample, the estimate of a causal effect may fail...

Academics : The Endeavour

Here’s an elegant little theorem I just learned about. Informally, A polynomial with few non-zero coefficients has few real roots. More precisely, If a polynomial has k non-zero coefficients, it can have at most 2k – 1 distinct real...

Academics : The Endeavour

Take a real number x and expand it as a continued fraction. Compute the geometric mean of the first n coefficients. Aleksandr Khinchin proved that for almost all real numbers x, as n ? ? the geometric means converge. Not only that, ...

Academics : The Endeavour

Binomial coefficients are hardly ever powers. That is, there are strong restrictions on when the equation has integer solutions for l ? 2. There are infinitely many solutions for k = 1 or 2. If k = 3 and l = 2, there is only one sol...

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