Consider the series

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Calculus of change and accumulation · Calculus of change and accumulation

Mathematical Methods for Physics, Ch Accumulation: integration and the techniques that make it possible

Problem

Consider the series ∑n=1∞n e−n2\displaystyle\sum_{n=1}^{\infty} n\,e^{-n^2}, which the ratio test already shows converges; here settle it again with the integral test, so the two methods can be compared. ((a)) Let f(x)=xe−x2f(x) = x e^{-x^2}. Show that ff is decreasing for every x≥1x \ge 1, and use the integral test to decide whether the series converges. Evaluate the improper integral ∫1∞xe−x2 dx\int_1^{\infty} x e^{-x^2}\,\mathrm{d}x. ((b)) Differentiate −12e−x2-\tfrac12 e^{-x^2} with respect to xx and confirm it reproduces the integrand xe−x2xe^{-x^2}, checking the antiderivative used in part (a) by the fundamental theorem of calculus.

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