The question asks:
Let $f:\mathbb{R}\longrightarrow\mathbb{R}$ be defined by $f(x)=e^x$
(a) Find the $n$th order Taylor polynomial about x = 0, and the corresponding remainder term.
I can quite comfortably find the $n$th order Taylor polynomial, just by expanding the Taylor series and then finding the pattern to get the Taylor polynomial which is:
However I get stuck when trying to find the remainder term. In my book it says the remainder term is equal to:
Could anyone help out?
The solution for the remainder term is:
The remainder of the Taylor expansion of a smooth function $f$ near $b$ is indeed $\dfrac{f^{(k+1)}(c)}{(k+1)!}(x-b)^{k+1}$, for some $c$ between $b$ and $x$. In particular, if $b=0$ and $f=\exp$, this will give you that the remainder is $\dfrac{e^c}{(k+1)!}x^{k+1}$, for some $c$ between $0$ and $x$.