A First Course in Complex Analysis · Problem 7.5.23

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

Mathematics — Complex Analysis · uniform-convergence-integral

Problem

Let

f_n(x) = n^2 x e^{-nx}

. (a) Show that

\lim_{n\to\infty} f_n(x) = 0

for all

x \geq 0

. Hint. Treat

x=0

as a special case; for

x>0

you can use L'H\^{o}pital's rule (Theorem A.0.11) — but remember that

n

is the variable, not

x

. (b) Find

\lim_{n\to\infty} \int_0^1 f_n(x)\,dx

. Hint. The answer is not 0. (c) Why doesn't your answer to part (b) violate Proposition 7.3.6?