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: Taylor series convert transcendental functions (like e^x, \sin x, \ln(1+x)) into polynomials. The key insight is that a function is completely determined by its derivatives at a single point (within the radius of convergence).

Common stuck point: A Taylor series may only equal the function within a certain radius of convergence. Outside this radius, the series diverges and is useless. For example, \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots only converges for -1 < x \leq 1.

Sense of Study hint: Build a table of n, f^(n)(a), and f^(n)(a)/n! for the first few terms to spot the pattern in the coefficients.

Worked Examples

Example 1

easy
Find the Maclaurin series for e^x up to the x^4 term.

Solution

  1. 1
    The Taylor series formula is f(x) = \sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n. Compute the derivatives of e^x at x=0.
  2. 2
    Since every derivative of e^x is e^x, we have f^{(n)}(0) = e^0 = 1 for all n.
  3. 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

Answer

e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \cdots
Every derivative of e^x is 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.

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).
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Background Knowledge

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

derivativedifferentiation rulesinfinite geometric seriesconvergence divergence