10.14 - Finding Taylor or Maclaurin Series for a Function

Definition of a Taylor Series

A Taylor series is an infinite series representation of a function, centered at

x = c

\boxed{f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!}(x-c)^n}

Expanded form:

f(x) = f(c) + f'(c)(x-c) + \frac{f''(c)}{2!}(x-c)^2 + \frac{f'''(c)}{3!}(x-c)^3 + \cdots

Key difference from Taylor polynomials: a Taylor series goes to infinity and actually equals the function, within its interval of convergence. A Taylor polynomial is just a finite approximation.

Taylor polynomials are partial sums of the Taylor series. The Taylor polynomial

P_n(x)

is the sum of all the series terms up through

(x-c)^n

. For

e^x

P_3(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!}

That’s the series

1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots

stopped after the

x^3

term. Raising the degree means adding more terms of the series, and inside the interval of convergence,

f(x) = \lim_{n \to \infty} P_n(x)

Watch the count:

P_n

stops at the power

n

, not after

n

terms. For

\sin(x)

P_5(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!}

has only three terms, because the even powers are missing.

Maclaurin Series (a Special Case)

A Maclaurin series is just a Taylor series centered at

x = 0

\boxed{f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n}

Expanded form:

f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots

Most problems involve Maclaurin series, centered at 0, rather than general Taylor series.

Four Essential Series to Memorize

You have to memorize these four fundamental Maclaurin series.

1. Exponential Function

\boxed{e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots}

Interval of convergence:

(-\infty, \infty)

2. Sine Function

\boxed{\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots}

Interval of convergence:

(-\infty, \infty)

Pattern: only odd powers, with alternating signs

3. Cosine Function

\boxed{\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots}

Interval of convergence:

(-\infty, \infty)

Pattern: only even powers, with alternating signs

4. Geometric Series Form

\boxed{\frac{1}{1+x} = \sum_{n=0}^{\infty} (-1)^n x^n = 1 - x + x^2 - x^3 + x^4 - \cdots}

Interval of convergence:

(-1, 1)

Memory Aid for Sine and Cosine

The alphabetical trick:

Cosine comes before Sine alphabetically

Even comes before Odd alphabetically

So: - Cosine goes with even powers:

x^0, x^2, x^4, x^6, \ldots

- Sine goes with odd powers:

x^1, x^3, x^5, x^7, \ldots

Deriving a Maclaurin Series from Scratch

Example: Deriving the Series for $e^x$

Step 1: Find all the derivatives

f(x) = e^x, \quad f'(x) = e^x, \quad f''(x) = e^x, \quad f'''(x) = e^x, \quad \ldots

Every derivative equals

e^x

Step 2: Evaluate at $x = 0$

f(0) = 1, \quad f'(0) = 1, \quad f''(0) = 1, \quad f'''(0) = 1, \quad \ldots

They all equal 1.

Step 3: Plug into the Maclaurin formula

e^x = \frac{1}{0!} + \frac{1}{1!}x + \frac{1}{2!}x^2 + \frac{1}{3!}x^3 + \cdots = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots = \sum_{n=0}^{\infty} \frac{x^n}{n!}

Example: A Taylor Series Centered at $x = 1$

Find the Taylor series for

f(x) = \frac{1}{x}

centered at

x = 1

Step 1: Find the derivatives and spot the pattern

f(x) = x^{-1}, \quad f'(x) = -x^{-2}, \quad f''(x) = 2x^{-3}, \quad f'''(x) = -6x^{-4}, \quad f^{(n)}(x) = (-1)^n n!\, x^{-(n+1)}

Step 2: Evaluate at

x = 1

f(1) = 1, \quad f'(1) = -1, \quad f''(1) = 2, \quad f'''(1) = -6, \quad f^{(n)}(1) = (-1)^n n!

Step 3: Plug into the Taylor formula. The

n!

cancels:

\frac{1}{x} = \sum_{n=0}^{\infty} \frac{(-1)^n n!}{n!}(x-1)^n = \sum_{n=0}^{\infty} (-1)^n (x-1)^n = 1 - (x-1) + (x-1)^2 - (x-1)^3 + \cdots

Check:

\frac{1}{x} = \frac{1}{1 + (x-1)}

, which is the

\frac{1}{1+x}

series with

x - 1

in place of

x

. It converges when

|x - 1| < 1

, so on

(0, 2)

The Geometric Series Connection

The series for

\frac{1}{1-x}

is actually a geometric series:

\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \cdots

This comes straight from the geometric series formula:

\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}

With

a = 1

and

r = x

\sum_{n=0}^{\infty} x^n = \frac{1}{1-x}

Related form:

\frac{1}{1+x} = \frac{1}{1-(-x)} = \sum_{n=0}^{\infty} (-x)^n = \sum_{n=0}^{\infty} (-1)^n x^n

Creating New Series by Substitution

Strategy: start with a known series and substitute a different expression for

x

Example 1: The Series for $\sin(x^2)$

Step 1: Write the known series for

\sin(x)

\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots

Step 2: Replace every

x

with

x^2

\sin(x^2) = x^2 - \frac{(x^2)^3}{3!} + \frac{(x^2)^5}{5!} - \frac{(x^2)^7}{7!} + \cdots

Step 3: Simplify the exponents

\sin(x^2) = x^2 - \frac{x^6}{3!} + \frac{x^{10}}{5!} - \frac{x^{14}}{7!} + \cdots

Step 4: Write it in summation notation

\sin(x^2) = \sum_{n=0}^{\infty} \frac{(-1)^n (x^2)^{2n+1}}{(2n+1)!} = \sum_{n=0}^{\infty} \frac{(-1)^n x^{4n+2}}{(2n+1)!}

Example 2: The Series for $e^{3x}$

Step 1: Write the known series for

e^x

e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots

Step 2: Replace every

x

with

3x

e^{3x} = 1 + 3x + \frac{(3x)^2}{2!} + \frac{(3x)^3}{3!} + \frac{(3x)^4}{4!} + \cdots

Step 3: Simplify

e^{3x} = 1 + 3x + \frac{9x^2}{2!} + \frac{27x^3}{3!} + \frac{81x^4}{4!} + \cdots

Step 4: Summation notation

e^{3x} = \sum_{n=0}^{\infty} \frac{(3x)^n}{n!} = \sum_{n=0}^{\infty} \frac{3^n x^n}{n!}

Example 3: The Series for $\cos(2x)$

Step 1: Start with

\cos(x)

\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots

Step 2: Replace

x

with

2x

\cos(2x) = 1 - \frac{(2x)^2}{2!} + \frac{(2x)^4}{4!} - \frac{(2x)^6}{6!} + \cdots

Step 3: Simplify

\cos(2x) = 1 - \frac{4x^2}{2!} + \frac{16x^4}{4!} - \frac{64x^6}{6!} + \cdots

Step 4: Summation notation

\cos(2x) = \sum_{n=0}^{\infty} \frac{(-1)^n (2x)^{2n}}{(2n)!} = \sum_{n=0}^{\infty} \frac{(-1)^n 2^{2n} x^{2n}}{(2n)!}

Creating New Series by Multiplication

Strategy: multiply the entire series by a function or expression.

Example 4: The Series for $x^2 e^x$

Step 1: Write the series for

e^x

e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots

Step 2: Multiply every term by

x^2

x^2 e^x = x^2 \left(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots\right) = x^2 + x^3 + \frac{x^4}{2!} + \frac{x^5}{3!} + \frac{x^6}{4!} + \cdots

Step 3: Summation notation

The general term from

e^x

\frac{x^n}{n!}

. Multiply it by

x^2

to get

\frac{x^{n+2}}{n!}

x^2 e^x = \sum_{n=0}^{\infty} \frac{x^{n+2}}{n!}

Example 5: The Series for $x\cos(x)$

Step 1: Start with

\cos(x)

\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots

Step 2: Multiply every term by

x

x\cos(x) = x - \frac{x^3}{2!} + \frac{x^5}{4!} - \frac{x^7}{6!} + \cdots

Step 3: Summation notation

x\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n)!}

First Few Terms vs. Summation Notation

Problems may ask for either format.

Option 1: The First Few Terms

“Find the first four terms of the Maclaurin series for

\sin(x^3)

.”

Answer: just write them out:

x^3 - \frac{x^9}{3!} + \frac{x^{15}}{5!} - \frac{x^{21}}{7!}

Option 2: Summation Notation

“Find the Maclaurin series for

\sin(x^3)

in summation notation.”

Answer:

\sum_{n=0}^{\infty} \frac{(-1)^n x^{6n+3}}{(2n+1)!}

Both are valid representations, so read the question carefully to know which one it wants.

Reading Information from a Taylor Series

A Taylor series stores every derivative of

f

at the center. The coefficient of

(x-c)^n

\frac{f^{(n)}(c)}{n!}

, so you can work backward:

f^{(n)}(c) = n! \cdot (\text{coefficient of } (x-c)^n)

The first few terms describe the graph at the center. The constant term is

f(c)

, the coefficient of

(x-c)

f'(c)

, and the coefficient of

(x-c)^2

\frac{f''(c)}{2}

. For

\cos(x) = 1 - \frac{x^2}{2!} + \cdots

, there’s no

x

term, so

f'(0) = 0

, and

f''(0) = 2! \cdot \left(-\frac{1}{2!}\right) = -1 < 0

. By the second derivative test,

\cos(x)

has a relative maximum at

x = 0

Example 6: A Derivative from the Series

Find

f^{(5)}(0)

for

f(x) = x^2 e^x

From Example 4,

x^2 e^x = x^2 + x^3 + \frac{x^4}{2!} + \frac{x^5}{3!} + \cdots

. The coefficient of

x^5

\frac{1}{3!}

, so

f^{(5)}(0) = 5! \cdot \frac{1}{3!} = \frac{120}{6} = 20

That’s much faster than taking five derivatives with the product rule.

Example 7: Recognizing a Series

Find the sum

1 - 2 + \frac{2^2}{2!} - \frac{2^3}{3!} + \frac{2^4}{4!} - \cdots

This is the series for

e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots

with

x = -2

, since the powers of

-2

give the alternating signs. The

e^x

series converges for every

x

, so the sum is

e^{-2} \approx 0.135

Summary Table of Essential Series

Function	Series (first few terms)	Summation Notation	Interval
$e^x$	$1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$	$\sum_{n=0}^{\infty} \frac{x^n}{n!}$	$(-\infty, \infty)$
$\sin(x)$	$x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$	$\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$	$(-\infty, \infty)$
$\cos(x)$	$1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$	$\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$	$(-\infty, \infty)$
$\frac{1}{1+x}$	$1 - x + x^2 - x^3 + \cdots$	$\sum_{n=0}^{\infty} (-1)^n x^n$	$(-1, 1)$
$\frac{1}{1-x}$	$1 + x + x^2 + x^3 + \cdots$	$\sum_{n=0}^{\infty} x^n$	$(-1, 1)$

Step-by-Step Strategy

To find a series for a new function:

Identify the parent function from the four memorized series

Determine the transformation. Is it a substitution, meaning you replace $x$ with something else? A multiplication, meaning you multiply the whole series by an expression? Both?

Apply the transformation to each term

Simplify the exponents and coefficients

Write the result in whichever form the problem wants

Common Transformations

Substitution Patterns

Original	Substitute	Result Function
$e^x$	$x \to -x$	$e^{-x}$
$e^x$	$x \to x^2$	$e^{x^2}$
$\sin(x)$	$x \to 3x$	$\sin(3x)$
$\cos(x)$	$x \to x^2$	$\cos(x^2)$
$\frac{1}{1+x}$	$x \to x^2$	$\frac{1}{1+x^2}$

Multiplication Patterns

Series	Multiply By	Result
$e^x$	$x$	$xe^x$
$\sin(x)$	$x^2$	$x^2\sin(x)$
$\cos(x)$	$x$	$x\cos(x)$

Key Takeaways

Taylor series = an infinite series that equals the function, within its convergence interval

Maclaurin series = a Taylor series centered at $x = 0$ , which is the most common case

Memorize four series: $e^x$ , $\sin(x)$ , $\cos(x)$ , and $\frac{1}{1+x}$

Sine goes with odd powers ( $x^1, x^3, x^5, \ldots$ ), and cosine goes with even powers ( $x^0, x^2, x^4, \ldots$ )

Alphabetical memory aid: Cosine/Even come before Sine/Odd

Substitution method: replace $x$ with a new expression in a known series

Multiplication method: multiply the entire series by an expression

Two answer formats: the first few terms, or summation notation

$\frac{1}{1-x}$ is a geometric series: $\sum_{n=0}^{\infty} x^n$ , which converges only on $(-1, 1)$

The exponential and trig series converge everywhere: $(-\infty, \infty)$

Power to a power rule: $(x^a)^b = x^{ab}$ when you’re simplifying exponents

Don’t forget the factorials in the denominators for $e^x$ , $\sin(x)$ , and $\cos(x)$

Back Next

10.14 - Finding Taylor or Maclaurin Series for a Function

Definition of a Taylor Series

Maclaurin Series (a Special Case)

Four Essential Series to Memorize

1. Exponential Function

2. Sine Function

3. Cosine Function

4. Geometric Series Form

Memory Aid for Sine and Cosine

Deriving a Maclaurin Series from Scratch

Example: Deriving the Series for exe^xex

Example: A Taylor Series Centered at x=1x = 1x=1

The Geometric Series Connection

Creating New Series by Substitution

Example 1: The Series for sin⁡(x2)\sin(x^2)sin(x2)

Example 2: The Series for e3xe^{3x}e3x

Example 3: The Series for cos⁡(2x)\cos(2x)cos(2x)

Creating New Series by Multiplication

Example 4: The Series for x2exx^2 e^xx2ex

Example 5: The Series for xcos⁡(x)x\cos(x)xcos(x)

First Few Terms vs. Summation Notation

Option 1: The First Few Terms

Option 2: Summation Notation

Reading Information from a Taylor Series

Example 6: A Derivative from the Series

Example 7: Recognizing a Series

Summary Table of Essential Series

Step-by-Step Strategy

Common Transformations

Substitution Patterns

Multiplication Patterns

Key Takeaways

Example: Deriving the Series for $e^x$

Example: A Taylor Series Centered at $x = 1$

Example 1: The Series for $\sin(x^2)$

Example 2: The Series for $e^{3x}$

Example 3: The Series for $\cos(2x)$

Example 4: The Series for $x^2 e^x$

Example 5: The Series for $x\cos(x)$