Practice Taylor Series in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

A representation of a function as an infinite sum of terms calculated from the function's derivatives at a single point: f(x)=∑n=0∞f(n)(a)n!(x−a)nf(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n
When a=0a = 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.

Showing a random 20 of 50 problems.

Example 1

hard
Evaluate lim⁡x→0sin⁡x−xx3\lim_{x \to 0} \frac{\sin x - x}{x^3} using Maclaurin series.

Example 2

medium
Find the Maclaurin series for x1+x2\frac{x}{1 + x^2} through the x5x^5 term.

Example 3

easy
True or false: the Taylor series of a polynomial of degree nn centered at aa has finitely many terms.

Example 4

challenge
Use Maclaurin series to show lim⁡x→0ex−1−xx2=12\lim_{x \to 0} \frac{e^x - 1 - x}{x^2} = \frac{1}{2}.

Example 5

medium
Use the Maclaurin series of cos⁡x\cos x and sin⁡x\sin x to verify Euler's identity in series form: show the real part of eixe^{ix} equals cos⁡x\cos x.

Example 6

easy
Find the first two nonzero terms of 11−x\frac{1}{1-x} as a Maclaurin series.

Example 7

medium
What is f(5)(0)f^{(5)}(0) if f(x)=sin⁡xf(x) = \sin x?

Example 8

easy
Write the Taylor series of f(x)=exf(x) = e^x about a=1a = 1 through the (x−1)2(x-1)^2 term.

Example 9

easy
Write the Maclaurin series for e−xe^{-x} through the x3x^3 term.

Example 10

challenge
Approximate ∫01e−x2 dx\int_0^1 e^{-x^2}\,dx using the first three series terms.

Example 11

medium
Find the Taylor series of ln⁡(1+x)\ln(1+x) through the x3x^3 term.

Example 12

easy
Write the Maclaurin series of sin⁡x\sin x through the x5x^5 term.

Example 13

medium
Approximate e0.2e^{0.2} using the Maclaurin series for exe^x through the x3x^3 term.

Example 14

challenge
Find the limit lim⁡x→0sin⁡x−xx3\lim_{x\to0}\frac{\sin x-x}{x^3} using series.

Example 15

hard
Find the Maclaurin series of f(x)=(1+x)1/2f(x) = (1+x)^{1/2} through the x3x^3 term.

Example 16

medium
Find the Maclaurin polynomial of degree 44 for f(x)=sec⁡xf(x) = \sec x.

Example 17

hard
Find the Maclaurin series for ln⁡(1+x)\ln(1+x) and state the interval of convergence.

Example 18

medium
Use the series sin⁡x≈x−x36\sin x\approx x-\frac{x^3}{6} to estimate sin⁡(0.5)\sin(0.5).

Example 19

medium
Find the Taylor series of ln⁡x\ln x about a=1a = 1 through the (x−1)3(x-1)^3 term.

Example 20

easy
Find the Maclaurin series of 11+x\frac{1}{1 + x} through the x3x^3 term.
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