Taylor Series

Definition

A Taylor series is a power series representation of a function $f(x)$ about a point $x=a$, constructed so that all derivatives of the series match those of $f$ at $a$.

Intuition

A Taylor series approximates a smooth function by matching its value and all its derivatives at a single point. The more terms included, the better the approximation over a wider region. The linear term recovers the familiar tangent-line approximation.

Formal Description

Writing $f(x)={\sum }_{n=0}^{\infty }{c}_n(x-a{)}^n$ and differentiating repeatedly at $x=a$ gives $c_n=f^{(n)}(a)/n!$, so

$$f(x)=\sum _{n=0}^{\infty }\frac{f^{(n)}(a)}{n!}(x-a{)}^n=f(a)+f^{\prime }(a)(x-a)+\frac{f^{\prime \prime }(a)}{2!}(x-a{)}^2+\frac{f^{\prime \prime \prime }(a)}{3!}(x-a{)}^3+\dots$$

The special case $a=0$ is called a Maclaurin series:

$$f(x)=f(0)+f^{\prime }(0)x+\frac{f^{\prime \prime }(0)}{2!}{x}^2+\frac{f^{\prime \prime \prime }(0)}{3!}{x}^3+\dots$$

Small-increment form. For small $ϵ$:

$$f(x+ϵ)=f(x)+f^{\prime }(x)ϵ+\frac{f^{\prime \prime }(x)}{2!}{ϵ}^2+\dots$$

Key Results — Common Taylor Series

All series below are centered at $0$.

Function	Taylor series	Radius of convergence
$e^x$	$\sum _{n=0}^{\infty }\frac{x^n}{n!}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots$	$\infty$
$\sin x$	$\sum _{n=0}^{\infty }\frac{(-1{)}^n{x}^{2n+1}}{(2n+1)!}=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$	$\infty$
$\cos x$	$\sum _{n=0}^{\infty }\frac{(-1{)}^n{x}^{2n}}{(2n)!}=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots$	$\infty$
$\frac{1}{1-x}$	$\sum _{n=0}^{\infty }{x}^n=1+x+x^2+x^3+\cdots$	$1$
$\frac{1}{1+x}$	$\sum _{n=0}^{\infty }(-1{)}^n{x}^n=1-x+x^2-x^3+\cdots$	$1$
$\ln (1+x)$	$\sum _{n=1}^{\infty }\frac{(-1{)}^{n+1}{x}^n}{n}=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots$	$1$
$\arctan x$	$\sum _{n=0}^{\infty }\frac{(-1{)}^n{x}^{2n+1}}{2n+1}=x-\frac{x^3}{3}+\frac{x^5}{5}-\cdots$	$1$
$(1+x{)}^p$	$1+px+\frac{p(p-1)}{2!}{x}^2+\frac{p(p-1)(p-2)}{3!}{x}^3+\cdots$	$1$
$\sqrt{1+x}$	$1+\frac{1}{2}x-\frac{1}{8}{x}^2+\cdots$	$1$

Derivation notes:

• $\cos x$ is obtained by differentiating the $\sin x$ series term-by-term.
• $\ln (1+x)$ is obtained by integrating the series for $\frac{1}{1+x}$, using $\ln 1=0$.
• $\arctan x$ is obtained by integrating the series for $\frac{1}{1+x^2}$ (substitute $x\mapsto x^2$ in $\frac{1}{1+x}$), using $\arctan 0=0$.

Remarkable special values:

$$\ln 2=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\dots ,\frac{\pi }{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\dots$$

Euler’s formula follows from substituting $x=i\theta$ into the exponential series and separating real and imaginary parts using the $\sin$ and $\cos$ series:

$$e^{i\theta }=\cos \theta +i\sin \theta .$$

Applications

Taylor series provide polynomial approximations to smooth functions. The linear approximation $f(x+ϵ)\approx f(x)+f^{\prime }(x)ϵ$ underlies L’Hôpital’s Rule.

Example. $\frac{1}{1+ϵ}\approx 1-ϵ$ for small $ϵ$ (linear Taylor approximation with $f(x)=1/x$, $x=1$).

Trade-offs

A Taylor series converges to $f(x)$ only within its radius of convergence and only for analytic functions. Functions like $e^{-1/x^2}$ are smooth but not analytic at $0$ — all Taylor coefficients vanish yet the function is nonzero, so the series fails to recover $f$.

Links

• Power Series
• Geometric Series
• L’Hôpital’s Rule
• Harmonic and p-Series — alternating harmonic series sums to $\ln 2$
• Complex Numbers — Euler’s formula $e^{i\theta }=\cos \theta +i\sin \theta$ follows from the exponential Taylor series