Taylor Series

Bartle & Donald’s book¹ on real analysis provides two theorems for Taylor series

Theorem 6.4.1: Taylor’s theorem

Let

• $n\in \mathbb{N}$
• $I:=[a,b]$
• $f:I\to \mathbb{R}$ such that
  • $f$ and its derivatives $(f^{\prime },f^{\prime \prime },\dots f^{(n)})$ are continuous on $I$
  • $f^{(n+1)}$ exists on $(a,b)$

If $x_0\in I$

Then

$$\begin{array}{rl}\forall x\in I,\exists c\in (x,x_0)& \ni \\ f(x)=& f(x_0)+f^{\prime }(x_0)(x-x_0)+\frac{f^{\prime \prime }(x_0)}{2!}(x-x_0{)}^2\\ & +\cdots +\frac{f^{(n)}(x_0)}{n!}(x-x_0{)}^n+\frac{f^{(n+1)}(c)}{(n+1)!}(x-x_0{)}^{n+1}\end{array}$$

Theorem 7.3.18: Taylor’s Theorem with the Remainder

Suppose

• $f^{\prime },\dots f^{(n)},f^{(n+1)}$ exists on $[a,b]$
• $f^{(n+1)}\in \mathcal{R}[a,b]$

Then

$$f(b)=f(a)+\frac{f^{\prime }(a)}{1!}(b-a)+\cdots +\frac{f^{(n)}(a)}{n!}(b-a{)}^n+R_n$$

Where remainder $R_n$ is given by

$$R_n=\frac{1}{n!}{\int }_a^b{f}^{(n+1)}(t)\cdot (b-t{)}^{n}dt$$

Extra: Maclaurun series

This theorem not included in Introduction to Real Analysis book but is important for common applications of Taylor series#6.4.1

Maclaurun series is a Taylor series expansion of a function centered at $x=0$. Below are some common Maclaurun series

$$\begin{array}{lllc}\frac{1}{1-x}& \sum _{k=0}^{\infty }{x}^k& 1+x+x^2+x^3+\dots & (-1,1)\\ \frac{1}{1+x^2}& \sum _{k=0}^{\infty }(-1{)}^k{x}^{2k}& 1-x^2+x^4-x^6+\dots & (-1,1)\\ e^x& \sum _{k=0}^{\infty }\frac{x^k}{k!}& 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\dots & (-\infty ,\infty )\\ \sin x& \sum _{k=0}^{\infty }\frac{(-1{)}^k{x}^{2k+1}}{(2k+1)!}& x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\dots & (-\infty ,\infty )\\ \cos x& \sum _{k=0}^{\infty }\frac{(-1{)}^k{x}^{2k}}{(2k)!}& 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\dots & (-\infty ,\infty )\\ \ln (1+x)& \sum _{k=1}^{\infty }\frac{(-1{)}^{k+1}{x}^k}{k}& x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\dots & (-1,1]\\ {\tan }^{-1}x& \sum _{k=0}^{\infty }\frac{(-1{)}^k{x}^{2k+1}}{2k+1}& x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\dots & [-1,1]\\ \sinh x& \sum _{k=0}^{\infty }\frac{x^{2k+1}}{(2k+1)!}& x+\frac{x^3}{3!}+\frac{x^5}{5!}+\frac{x^7}{7!}+\dots & (-\infty ,\infty )\\ \cosh x& \sum _{k=0}^{\infty }\frac{x^{2k}}{(2k)!}& 1+\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+\dots & (-\infty ,\infty )\\ (1+x{)}^m& \sum _{k=0}^{\infty }\left(\genfrac{}{}{0ex}{}{m}{k}\right)x^k& 1+mx+\frac{m(m-1)}{2!}{x}^2+\dots & (-1,1{)}^{\ast }\end{array}$$

Footnotes

1. Bartle, Robert G., and Donald R. Sherbert. Introduction to Real Analysis. ↩