Taylor & Maclaurin Series Cheat Sheet

Key Facts

• The Taylor series for $f(x)$ centered at $x=a$ is $\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$.
• The Maclaurin series is the Taylor series centered at $a=0$, 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 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 exponential series is $e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}$ for all real $x$.
• The sine and cosine series are $\sin x=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!}$ and $\cos x=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!}$.
• The Lagrange error bound is $|R_n(x)|\le \frac{M}{(n+1)!}|x-a|^{n+1}$ when $|f^{(n+1)}(t)|\le M$ between $a$ and $x$.
• The ratio test checks convergence using $L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$, where the series converges if $L<1$ and diverges if $L>1$.

Vocabulary

Taylor Series

An infinite power series that represents a function using its derivatives at a center $a$.

Maclaurin Series

A Taylor series centered at $a=0$.

Taylor Polynomial

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

Radius of Convergence

The distance $R$ from the center where a power series converges for $|x-a|<R$.

Interval of Convergence

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

Remainder

The error term $R_n(x)=f(x)-P_n(x)$ between the actual function value and the Taylor polynomial approximation.