Taylor Series Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Taylor Series.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A representation of a function as an infinite sum of terms calculated from the function's derivatives at a single point:

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

When

a = 0

, it's called a Maclaurin series.

Approximate any smooth function with a polynomial by matching the function's value, slope, curvature, and all higher derivatives at a single point. The more terms you include, the wider the region where the polynomial closely matches the function. It's like fitting a polynomial glove onto the function's hand.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A Taylor series rebuilds a function as an infinite polynomial whose derivatives at $a$ match the function's.

Common stuck point: The procedure for taylor series is the easy part; the trap is forgetting the $n!$ in the denominator. Asking "Am I building an infinite polynomial whose successive derivatives at one center match the function's?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I building an infinite polynomial whose successive derivatives at one center match the function's?

Worked Examples

Example 1

easy

Find the Maclaurin series for

e^x

up to the

x^4

term.

Answer

e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \cdots

First step

The Taylor series formula is

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

. Compute the derivatives of

e^x

x=0

Full solution

2
Since every derivative of $e^x$ is $e^x$ , we have $f^{(n)}(0) = e^0 = 1$ for all $n$ .
3
Substitute into the formula: $e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!} = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots$

Every derivative of

e^x

e^x

, so all coefficients are

1/n!

. Converges for all

x

Example 2

hard

Find the Maclaurin series for

\ln(1+x)

and state the interval of convergence.

Example 3

medium

Approximate

e^{0.2}

using the Maclaurin series for

e^x

through the

x^3

term.

Example 4

medium

Find the Taylor series of

\ln x

about

a = 1

through the

(x-1)^3

term.

Example 5

medium

Use the Maclaurin series of

\cos x

and

\sin x

to verify Euler's identity in series form: show the real part of

e^{ix}

equals

\cos x

Example 6

hard

Evaluate

\lim_{x \to 0} \frac{\sin x - x}{x^3}

using Maclaurin series.

Example 7

hard

Use the first non-zero error term of the Maclaurin series for

\cos x

to bound the error in approximating

\cos(0.5)

1 - \frac{x^2}{2}

Example 8

hard

Find the Maclaurin series of

\arctan x

and use it to write

\pi

as a series.

Example 9

hard

Compute the Maclaurin series of

f(x) = e^x \cos x

through the

x^3

term.

Example 10

challenge

Use Maclaurin series to show

\lim_{x \to 0} \frac{e^x - 1 - x}{x^2} = \frac{1}{2}

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Write the first four non-zero Maclaurin terms for

\sin x

Example 2

medium

Use three terms of the Maclaurin series for

\cos x

to approximate

\cos(0.1)

Example 3

easy

Write the Maclaurin series of

e^x

Example 4

easy

Write the Maclaurin series of

\sin x

through the

x^5

term.

Example 5

easy

Write the Maclaurin series of

\cos x

through the

x^4

term.

Example 6

easy

What is the coefficient of

(x-a)^n

in a Taylor series?

Example 7

easy

Find the first two nonzero terms of

\frac{1}{1-x}

as a Maclaurin series.

Example 8

easy

What is the constant term of the Taylor series of

f

about

a

Example 9

easy

The first-degree Taylor polynomial of

f

a

is what?

Example 10

easy

1+x+x^2

the exact value or an approximation of

\frac{1}{1-x}

Example 11

medium

Find the Taylor series of

\ln(1+x)

through the

x^3

term.

Example 12

medium

Use

e^x

's series to approximate

e^{0.1}

to three terms.

Example 13

medium

Find the Maclaurin series of

e^{-x^2}

through the

x^4

term.

Example 14

medium

Find the Taylor coefficient of

x^2

for

f(x)=\cos x

0

Example 15

medium

Multiply series to find the

x^2

coefficient of

e^x\sin x

Example 16

medium

Differentiate the series of

\sin x

term by term to get

\cos x

Example 17

medium

Find the Taylor series of

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

about

a=1

through

(x-1)^2

Example 18

medium

Use the series

\sin x\approx x-\frac{x^3}{6}

to estimate

\sin(0.5)

Example 19

medium

Find the Maclaurin series of

\frac{1}{1+x^2}

through the

x^4

term.

Example 20

challenge

Find the limit

\lim_{x\to0}\frac{\sin x-x}{x^3}

using series.

Example 21

challenge

Approximate

\int_0^1 e^{-x^2}\,dx

using the first three series terms.

Example 22

challenge

Show why

\sin x\approx x-\frac{x^3}{6}

has error bounded by

\frac{x^5}{120}

Example 23

easy

Write the Maclaurin series for

e^{-x}

through the

x^3

term.

Example 24

easy

What is the Maclaurin polynomial of degree

2

for

f(x) = \sin x

Example 25

easy

Write the Taylor series of

f(x) = e^x

about

a = 1

through the

(x-1)^2

term.

Example 26

easy

Find the Maclaurin series of

\frac{1}{1 + x}

through the

x^3

term.

Example 27

easy

Find the Maclaurin series of

\sinh x

through the

x^5

term.

Example 28

medium

Use the Maclaurin series of

\sin x

to estimate

\int_0^1 \sin(x^2)\,dx

to four decimal places.

Example 29

medium

Find the Maclaurin series for

f(x) = x \cos x

through the

x^5

term.

Example 30

medium

What is

f^{(5)}(0)

f(x) = \sin x

Example 31

medium

Find the Maclaurin series for

\frac{x}{1 + x^2}

through the

x^5

term.

Example 32

medium

Find the Maclaurin polynomial of degree

4

for

f(x) = \sec x

Example 33

hard

Find the Maclaurin series of

f(x) = (1+x)^{1/2}

through the

x^3

term.

Example 34

hard

What is

f^{(10)}(0)

f(x) = x^2 \sin x

Example 35

hard

Approximate

\ln(1.1)

using three non-zero terms of the Maclaurin series for

\ln(1+x)

Example 36

hard

Find the Maclaurin series of

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

(define

f(0) = 1

) through the

x^4

term.

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Related Concepts

Derivative Differentiation Rules Infinite Geometric Series Convergence and Divergence Power Series

Background Knowledge

These ideas may be useful before you work through the harder examples.

derivativedifferentiation rulesinfinite geometric seriesconvergence divergence