Basic Analysis I — Real Analysis · Problem 5.4.11

The Riemann Integral — Jiri Lebl

Mathematics — Real Analysis · smooth-bump-function

Problem

(requires §4.3) Since

(e^x)' = e^x

, it is easy to see that

e^x

is infinitely differentiable (has derivatives of all orders). Define the function

f : \mathbb{R} \to \mathbb{R}

f(x) := \begin{cases} e^{-1/x} & \text{if } x > 0, \\ 0 & \text{if } x \leq 0. \end{cases}

a) Prove that for every

m \in \mathbb{N}

\lim_{x\to 0^+} \frac{e^{-1/x}}{x^m} = 0.

b) Prove that

f

is infinitely differentiable. c) Compute the Taylor series for

f

at the origin, that is,

\sum_{k=0}^\infty \frac{f^{(k)}(0)}{k!} x^k.

Show that it converges, but show that it does not converge to

f(x)

for any given

x > 0