Taylor Series — Question 1

*Question 1 Construct the Maclaurin series for $e^x$ from its derivatives. Prove that it converges to $e^x$ for every real $x$, and give a uniform Lagrange-remainder bound for the degree-$N$ polynomial on $|x|\le1$.

*Question 1 – Solution

Every derivative of $f(x)=e^x$ is $e^x$, so $f^{(n)}(0)=1$. Hence $$P_N(x)=\sum_{n=0}^{N}\frac{x^n}{n!},\qquad \boxed{e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}}.$$

Taylor’s theorem gives $$R_N(x)=\frac{e^c x^{N+1}}{(N+1)!}$$ for some $c$ between $0$ and $x$. If $|x|\le1$, then $c\le1$, so $$\boxed{|R_N(x)|\le\frac{e|x|^{N+1}}{(N+1)!}\le\frac{e}{(N+1)!}}.$$

For any fixed real $x$, $e^c\le e^{|x|}$ and $e^{|x|}|x|^{N+1}/(N+1)!\to0$ by the Ratio Test. Thus $R_N(x)\to0$ for every $x$, proving the series represents $e^x$ on all of $\mathbb R$.