Taylor Series

Level 1 - Math I (Physics) topic page in Power Series.

Principle

A Taylor series is the infinite power series built from all derivatives of a function at a centre. When it converges to the function, it gives a local polynomial representation with infinitely many terms.

Taylor series

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

The special case \(c=0\) is called a Maclaurin series. Many common functions in physics are used through their Maclaurin series near zero.

Notation

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

nth derivative of f at the centre c

\(n!\)

factorial of n

\(c\)

centre of the Taylor series

\(x-c\)

small displacement from the centre

\(\approx\)

approximately equal, often after truncating the series

A Taylor series may converge without equalling the original function everywhere. In standard Level 1 examples such as \(e^x\), \(\sin x\), \(\cos x\), and \((1-x)^{-1}\), it does represent the function on its convergence interval.

Method

Step 1: Compute derivative values

Make a table of \(f(c), f'(c), f''(c)\), and so on. Look for a repeating pattern.

Step 2: Substitute into the Taylor formula

Use the coefficient formula term by term:

General coefficient

\[a_n=\frac{f^{(n)}(c)}{n!}\]

Power of displacement

\[(x-c)^n\]

Series term

\[\frac{f^{(n)}(c)}{n!}(x-c)^n\]

Sum all terms

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

Step 3: State where it is valid

Find or quote the convergence interval. A Taylor formula without its validity range can be misleading.

Rules

Maclaurin series for exponential

\[e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}\]

Maclaurin series for sine

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

Maclaurin series for cosine

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

Geometric series

\[\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n\quad(|x|\lt1)\]

Examples

Question

Write the first four non-zero terms of the Maclaurin series for \(e^x\).

Answer

Use

\[e^x=\sum x^n/n!\]

The first four non-zero terms are

\[1+x+x^2/2!+x^3/3!\]

Checks

Taylor series are infinite; Taylor polynomials are finite truncations.
Always state the centre.
Radians are required for the standard sine and cosine series.
Convergence to the function is an additional statement, not just notation.

Taylor Polynomials

Taylor Theorem