Step By Step Calculus » 17.6 - Taylor Series

Synopsis
Taylor series representation of a function f(x)f(x) at x=ax=a has the following form
\displaystyle \sum_{i=0}^{\infty}\frac{f^{(i)}(a)}{i!}(x-a)^i
\displaystyle \sum_{i=0}^{\infty}\frac{f^{(i)}(a)}{i!}(x-a)^i
and the partial sums of a Taylor series is called Taylor polynomial where
\displaystyle T_n(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k
\displaystyle T_n(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k
is the n^{\textrm{th}}n^{\textrm{th}}-degree Taylor polynomial of ff at aa. f(x)f(x) can be expressed as a sum of T_n(x)T_n(x) and a remainder R_n(x)R_n(x). This remainder term of Taylor series can be expressed in Lagrange’s form which states: If f^{(n)}(x)f^{(n)}(x) is differentiable in the open interval between aa and xx where |x-a|<R|x-a|<R, then there exists a number cc between aa and xx such that
\displaystyle R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}.
\displaystyle R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}.
A corollary of the above Lagrange’s form is the following Taylor inequality.
 \displaystyle |R_n(x)|\leq \frac{M}{(n+1)!}|x-a|^{n+1}\displaystyle |R_n(x)|\leq \frac{M}{(n+1)!}|x-a|^{n+1}       where \displaystyle \left. M=\textrm{LUB}\{f^{(n+1)}(c) \right| c \displaystyle \left. M=\textrm{LUB}\{f^{(n+1)}(c) \right| c is between \displaystyle x\displaystyle x and \displaystyle a\} \displaystyle a\}
Taylor series of a function f(x)f(x) converges to f(x)f(x) if
 \displaystyle \lim_{n\to\infty} T_n(x)=f(x) \displaystyle \lim_{n\to\infty} T_n(x)=f(x)        or in other words if \displaystyle \lim_{n\to\infty}R_n(x)=0 \displaystyle \lim_{n\to\infty}R_n(x)=0
Taylor series of a function ff at a=0a=0 is called Maclaurin series. Maclaurin series of the function (1+x)^m(1+x)^m for m\in\mathbb{R}m\in\mathbb{R} is known as Binomial series.