Taylor and Maclaurin Series — AP® Calculus BC Notes

A Taylor polynomial has a last term. If you never stop adding terms, you get a power series, the Taylor series of the function. For the functions in this course (the four series below and every series built from them), that series does more than approximate: on its interval of convergence it equals the function. That is a property of these functions, not of every function: a few functions have a Taylor series that converges to something else, and the course does not use them.

From polynomials to a series

The Taylor series of $f$ about $x = a$ is

$\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \cdots$

About $a = 0$ it is called the Maclaurin series. The Taylor polynomial $P_n$ is its partial sum through the $(x - a)^n$ term. So the questions of the last three topics fit together: $P_n(x)$ is a partial sum, the error bounds measure how far a partial sum is from the sum, and the interval of convergence says where the partial sums settle at all.

The graph of y = e to the x from x = -3 to 2.5 with the first three Taylor polynomials about 0: the line 1 + x, the parabola 1 + x + x squared over 2, and the cubic that adds x cubed over 6. Each new term brings the polynomial closer to e to the x over a wider stretch; at x = 2 their values are 3, 5 and about 6.333, approaching e squared, about 7.389. — **Figure 1.** The partial sums of the Maclaurin series for $e^x$ close in on the curve. LevlPrep original diagram.

For $e^x$ , every derivative at 0 is 1, so the Maclaurin series is $\sum \frac{x^n}{n!}$ . The figure shows its first partial sums closing in on $e^x$ . The ratio test gives $\frac{|x|}{n + 1} \to 0$ for every $x$ , so the series converges everywhere, and it converges to $e^x$ .

Four series to know by heart

Function	Maclaurin series	General term	Converges to the function for
$\frac{1}{1 - x}$	$1 + x + x^2 + x^3 + \cdots$	$x^n$	$-1 < x < 1$
$e^x$	$1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$	$\frac{x^n}{n!}$	every $x$
$\sin x$	$x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$	$\frac{(-1)^n x^{2n+1}}{(2n+1)!}$	every $x$
$\cos x$	$1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$	$\frac{(-1)^n x^{2n}}{(2n)!}$	every $x$

Patterns help you remember them. $\sin x$ is odd, so only odd powers appear; $\cos x$ is even, so only even powers. Both alternate in sign because their derivatives cycle through $\sin, \cos, -\sin, -\cos$ . The series for $\frac{1}{1 - x}$ is just the geometric series, which is why it works only for $|x| < 1$ .

Building a Taylor series from derivatives

Worked example 1. Find the Taylor series for $f(x) = e^x$ about $x = 2$ .

Every derivative of $e^x$ is $e^x$ , so $f^{(n)}(2) = e^2$ for every $n$ .

The coefficients are $\frac{e^2}{n!}$ , and the series is $\sum_{n=0}^{\infty} \frac{e^2 (x - 2)^n}{n!}$ $= e^2$ $+ e^2(x - 2)$ $+ \frac{e^2}{2!}(x - 2)^2$ $+ \cdots$

When a question gives a formula for $f^{(n)}(a)$ instead of the function, the same recipe applies: divide by $n!$ and simplify. If $f^{(n)}(0) = \frac{n!}{3^n}$ , the coefficients are $\frac{1}{3^n}$ and the series is the geometric series $\sum \left(\frac{x}{3}\right)^n$ .

Reading coefficients backward

If a function is given by a power series $\sum c_n (x - a)^n$ with a positive radius, then that series is its Taylor series, so $c_n = \frac{f^{(n)}(a)}{n!}$ and

$f^{(n)}(a) = n! \cdot c_n.$

Worked example 2. $g(x)$ $= \sum_{n=0}^{\infty} \frac{(-1)^n x^n}{n + 2}$ . Find $g''(0)$ .

The coefficient of $x^2$ is $c_2$ $= \frac{(-1)^2}{2 + 2}$ $= \frac{1}{4}$ , so $g''(0) = 2! \cdot \frac{1}{4} = \frac{1}{2}$ .

Using known series to find sums and limits

Worked example 3. Find $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n + 1)!}\left(\frac{\pi}{6}\right)^{2n+1}$ .

This is the series for $\sin x$ with $x = \frac{\pi}{6}$ , so the sum is $\sin\frac{\pi}{6} = \frac{1}{2}$ .

Worked example 4. Find $\lim_{x \to 0} \frac{1 - \cos x}{x^2}$ .

$1 - \cos x$ $= \frac{x^2}{2!} - \frac{x^4}{4!} + \cdots$ , so $\frac{1 - \cos x}{x^2}$ $= \frac{1}{2} - \frac{x^2}{24}$ $+ \cdots \to \frac{1}{2}$ .

Series or polynomial?

When a question asks for a Taylor polynomial, write a finite sum that ends at the stated degree. When it asks for a series, write the first few terms followed by " $+ \cdots$ ", or use sigma notation with the general term. Mixing the two, a polynomial with " $+ \cdots$ " or a series with no general term when one is asked for, costs points on the exam.