[Actually, I made a slight typo in the previous e-mail. But this one is slicker since it doesn't require differentiation of log.] log(x^2 + 1) = log(x + i) + log(x - i), so it suffices to show that the (4n + 2)th derivatives of log(x + i) are all pure-imaginary when x = 1. Let x = 1 + y, so we are interested in the Taylor series of: log(y + (1 + i)) about the point y = 0. We want to show that the (4n + 2)th coefficients are all pure-imaginary. This follows from the claim that the coefficient in y^n of the series expansion of log(y + a) is [a real multiple of] a^-n, which is true by dimensional analysis. Sincerely, Adam P. Goucher

Sent: Thursday, November 12, 2015 at 3:24 PM From: "Erich Friedman" <erichfriedman68@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] ln(x^2+1)

this morning i found that for f(x)=ln(x^2+1), that the (4n+2)th derivatives all vanish at x=1. it took me an hour to figure out why, using geometric series, but surely there must be an easier way to see this. does anyone have an elegant proof of this?

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