Taylor Series Math Example 3

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Example 3

easy
Write the first four non-zero Maclaurin terms for sin⁡x\sin x.

Solution

  1. 1
    Odd-function: only odd powers survive.
  2. 2
    sin⁡x=x−x36+x5120−x75040+⋯\sin x = x - \frac{x^3}{6} + \frac{x^5}{120} - \frac{x^7}{5040} + \cdots

Answer

x−x36+x5120−x75040+⋯x - \frac{x^3}{6} + \frac{x^5}{120} - \frac{x^7}{5040} + \cdots
Only odd-power terms appear. The series converges for all xx.

About Taylor Series

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.

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More Taylor Series Examples

Example 1 easy

Find the Maclaurin series for [formula] up to the [formula] term.

Example 2 hard

Find the Maclaurin series for [formula] and state the interval of convergence.

Example 4 medium

Use three terms of the Maclaurin series for [formula] to approximate [formula].

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