---
# System prepended metadata

title: 數學歸納法的練習
tags: [Advanced Calculus]

---

###### tags: `Advanced Calculus` 
# 數學歸納法的練習
Let $$f(x) = \left\{ \begin{array}{ll} \frac{1}{e^{\frac{1}{x}}} & if\; x > 0;
\\ 0 & if\; x \leq 0.\end{array} \right. $$ 
Show that for each $$n=0, 1, 2, 3 ,\dots, $$ there is a polynomial $$P_n (x)$$ such that
 $$ f^{\left(n\right)}(x) = \left\{ \begin{array}{ll} P_n (\frac{1}{x})\frac{1}{e^{\frac{1}{x}}} & if\; x > 0;
\\ 0 & if\; x \leq 0.\end{array} \right. $$
Does $f$ equal its Maclaurin series?