Taylor and Maclaurin Series

Taylor and Maclaurin Series

Taylor Expansion Theorem:

Let $f$ be analytic in a domain $D$ and $z_0$ be a point in $D$. Then $f$ can be expanded in a power series

\[f(z) = f(z_0) + f'(z_0)(z - z_0) + \frac{f''(z_0)}{2!}(z - z_0)^2 + \cdots,\]

valid in all circles $|z - z_0| < r$ containing only points of $D$.

The expansion

\[f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(z_0)}{n!} (z - z_0)^n\]

is called the Taylor series of $f$ about $z_0$.

The special case in which $z_0$ = 0

\[f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} z^n\]

is called the Maclaurin series of f.

The circle $|z - z_0| < $ in which the Taylor series converges to the function is called the circle of convergence for the Taylor series.

Every power series representation of (or, Taylor series for) an entire function has an infinite radius of convergence.

A theorem about the uniqueness of Taylor series as power series expansions: if $f$ has a power series expansion about a point $z_0$ with nonzero radius of convergence, it must be the Taylor series about $z_0$.

A theorem about radius of convergence: The radius of convergence of the Taylor series for a function $f(z)$ about a point $z_0$ is the distance from $z_0$ to the nearest singularity of $f(z)$.

Here are some important and useful Maclaurin series that we can often use to find those of other functions:

\[e^z = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots = \sum_{n=0}^{\infty} \frac{1}{n!}z^n, |z| < \infty,\] \[\sin{z} = z - \frac{z^3}{3!} + \frac{z^5}{5!} - \frac{z^7}{7!} + \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}z^{2n+1}, |z| < \infty,\] \[\cos{z} = z - \frac{z^2}{2!} + \frac{z^4}{4!} - \frac{z^6}{6!} + \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!}z^{2n}, |z| < \infty,\] \[\frac{1}{1 - z} = 1 + z + z^2 + z^3 + \cdots = \sum_{n = 0}^{\infty} z^n, |z| < 1.\]