A First Course in Complex Analysis · Problem 7.5.36

Taylor and Laurent Series — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · uniform-convergence-improper-integral

Problem

Define the functions fn:R≥0→Rf_n : \mathbb{R}_{\geq 0} \to \mathbb{R} via fn(t)=1ne−tnf_n(t) = \frac{1}{n}e^{-\frac{t}{n}}, for n≥1n \geq 1. (a) Show that the maximum of fn(t)f_n(t) is 1n\frac{1}{n}. (b) Show that fn(t)f_n(t) converges uniformly to the zero function on R≥0\mathbb{R}_{\geq 0}. (c) Show that ∫0∞fn(t) dt\int_0^\infty f_n(t)\,dt does not converge to 0 as n→∞n \to \infty. (d) Why doesn't this contradict Proposition 7.3.6?
Intuitive explanation

These functions form a family that shrinks in height but spreads out in width—a classic example of how pointwise and uniform convergence can behave differently from integral behavior. Part (a) is just calculus; parts (b) and (c) show that even though fn→0f_n \to 0 uniformly everywhere, the total area under each curve stays constant, revealing why you can't always swap limits and integrals. Part (d) asks you to identify which hypothesis of the relevant convergence theorem fails, teaching you that uniform convergence alone isn't enough to guarantee integrals converge.

Intuitive explanation

These functions form a family that shrinks in height but spreads out in width—a classic example of how pointwise and uniform convergence can behave differently from integral behavior. Part (a) is just calculus; parts (b) and (c) show that even though fn→0f_n \to 0 uniformly everywhere, the total area under each curve stays constant, revealing why you can't always swap limits and integrals. Part (d) asks you to identify which hypothesis of the relevant convergence theorem fails, teaching you that uniform convergence alone isn't enough to guarantee integrals converge.

Related theory

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