Taylor and Maclaurin Series Reference Cheat Sheet

Key Facts

• The Taylor series for $f(x)$ centered at $a$ is $\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$.
• The Maclaurin series is the Taylor series centered at $0$, so $f(x)=\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$ when the series converges to $f(x)$.
• The $n$th Taylor polynomial centered at $a$ is $P_n(x)=\sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^k$.
• The geometric series formula is $\frac{1}{1-x}=\sum_{n=0}^{\infty} x^n$ for $|x|<1$.
• The Maclaurin series for $e^x$ is $e^x=\sum_{n=0}^{\infty} \frac{x^n}{n!}$ for all real $x$.
• The Maclaurin series for $\sin x$ is $\sin x=\sum_{n=0}^{\infty} (-1)^n\frac{x^{2n+1}}{(2n+1)!}$ for all real $x$.
• The Maclaurin series for $\cos x$ is $\cos x=\sum_{n=0}^{\infty} (-1)^n\frac{x^{2n}}{(2n)!}$ for all real $x$.
• Lagrange's remainder formula is $R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$ for some $c$ between $a$ and $x$.

Vocabulary

Taylor series

An infinite polynomial representation of a function centered at $a$, using the values of the function's derivatives at $a$.

Maclaurin series

A Taylor series centered at $0$, written using powers of $x$ and derivatives evaluated at $0$.

Taylor polynomial

A finite polynomial $P_n(x)$ made from the first $n+1$ terms of a Taylor series.

Remainder

The error $R_n(x)=f(x)-P_n(x)$ between a function and its $n$th Taylor polynomial approximation.

Radius of convergence

The distance $R$ from the center of a power series within which the series converges.

Interval of convergence

The set of $x$-values for which a power series converges, including any endpoints that work.