Taylor Series

Introduction

Taylor and Maclaurin series allow us to approximate functions using polynomials. These series are particularly useful in physics when we need to simplify complex functions or analyze the behavior of systems near a specific point. In this section, we'll introduce both series and demonstrate their applications with several examples.

The Maclaurin Series

The Maclaurin series is a special case of the Taylor series, where the expansion is centered at $x = 0$. If a function $f(x)$ has derivatives of all orders at $x = 0$, then it can be expressed as an infinite series:

Maclaurin Series

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots

where $f^{(n)}(0)=\frac{d^n f}{dx^n}\bigg|_{x=0}$ denotes the $n$-th derivative of $f$ evaluated at $x = 0$, and $n!$ is the factorial of $n$.

The Maclaurin series provides a polynomial approximation of the function near $x = 0$. For many functions, this series converges to the function value within a certain radius of convergence.

The Taylor Series

The Taylor series is a more general expansion that allows us to approximate a function around any point $a$. If a function $f(x)$ has derivatives of all orders at $x = a$, then it can be expressed as:

Taylor Series

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots

where $f^{(n)}(a)=\frac{d^n f}{dx^n}\bigg|_{x=a}$ is the $n$-th derivative of $f$ evaluated at $x = a$. Notice that when $a = 0$, the Taylor series reduces to the Maclaurin series.

The Taylor series is useful when we want to approximate a function near a specific point other than the origin.

Example: Taylor Series for $f(x) = \sqrt{x}$ centered at $x = 4$

To find the Taylor series for $f(x) = \sqrt{x}$ centered at $x = 4$, we need to compute the derivatives and evaluate them at $x = 4$:

$f(x) = x^{1/2}$, so $f(4) = \sqrt{4} = 2$
$f'(x) = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}$, so $f'(4) = \frac{1}{2\sqrt{4}} = \frac{1}{4}$
$f''(x) = -\frac{1}{4}x^{-3/2} = -\frac{1}{4x^{3/2}}$, so $f''(4) = -\frac{1}{4(4^{3/2})} = -\frac{1}{4 \cdot 8} = -\frac{1}{32}$
$f'''(x) = \frac{3}{8}x^{-5/2} = \frac{3}{8x^{5/2}}$, so $f'''(4) = \frac{3}{8(4^{5/2})} = \frac{3}{8 \cdot 32} = \frac{3}{256}$

Substituting into the Taylor series formula with $a = 4$:

\begin{align*} \sqrt{x} &= f(4) + f'(4)(x-4) + \frac{f''(4)}{2!}(x-4)^2 + \frac{f'''(4)}{3!}(x-4)^3 + \cdots \\ &= 2 + \frac{1}{4}(x-4) - \frac{1}{32 \cdot 2!}(x-4)^2 + \frac{3}{256 \cdot 3!}(x-4)^3 + \cdots \\ &= 2 + \frac{1}{4}(x-4) - \frac{1}{64}(x-4)^2 + \frac{1}{512}(x-4)^3 + \cdots \end{align*}

Application: We can use this series to approximate $\sqrt{4.1}$. Since $x = 4.1$ and $x - 4 = 0.1$:

\begin{align*} \sqrt{4.1} &\approx 2 + \frac{1}{4}(0.1) - \frac{1}{64}(0.1)^2 + \frac{1}{512}(0.1)^3 \\ &= 2 + 0.025 - \frac{0.01}{64} + \frac{0.001}{512} \\ &= 2 + 0.025 - 0.00015625 + 0.00000195 \\ &\approx 2.0248 \end{align*}

The exact value is $\sqrt{4.1} \approx 2.02485$, so our three-term approximation is accurate to within 0.002%. This demonstrates how the Taylor series centered at $x = 4$ provides an excellent approximation for values near $x = 4$.

Example: Maclaurin series for $f(x) = \sin(x)$

To derive the Maclaurin series for $f(x) = \sin(x)$, we need to compute the derivatives of $\sin(x)$ and evaluate them at $x = 0$:

$f(0) = \sin(0) = 0$
$f'(x) = \cos(x)$, so $f'(0) = \cos(0) = 1$
$f''(x) = -\sin(x)$, so $f''(0) = -\sin(0) = 0$
$f'''(x) = -\cos(x)$, so $f'''(0) = -\cos(0) = -1$
$f^{(4)}(x) = \sin(x)$, so $f^{(4)}(0) = \sin(0) = 0$

The pattern repeats every four derivatives. Substituting into the Maclaurin series formula:

Maclaurin Series for $\sin(x)$:

\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}

This series converges for all real values of $x$. Notice that only odd powers of $x$ appear, which reflects the fact that $\sin(x)$ is an odd function: $\sin(-x) = -\sin(x)$.

Example: Maclaurin series for $f(x) = \cos(x)$

We follow the same method used in the example above. We start by computing the derivatives of $\cos(x)$ and evaluate them at $x = 0$:

$f(0) = \cos(0) = 1$
$f'(x) = -\sin(x)$, so $f'(0) = -\sin(0) = 0$
$f''(x) = -\cos(x)$, so $f''(0) = -\cos(0) = -1$
$f'''(x) = \sin(x)$, so $f'''(0) = \sin(0) = 0$
$f^{(4)}(x) = \cos(x)$, so $f^{(4)}(0) = \cos(0) = 1$

The pattern also repeats every four derivatives. The Maclaurin series is:

Maclaurin Series for $\cos(x)$:

\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}

This series also converges for all real $x$. Only even powers of $x$ appear, reflecting that $\cos(x)$ is an even function: $\cos(-x) = \cos(x)$.

Example: Maclaurin Series for $e^x$

For the exponential function $f(x) = e^x$, all derivatives are equal to $e^x$:

$f(0) = e^0 = 1$
$f'(0) = e^0 = 1$
$f''(0) = e^0 = 1$
$f^{(n)}(0) = 1$ for all $n$

Substituting into the Maclaurin series:

Maclaurin Series for $e^x$:

e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots = \sum_{n=0}^{\infty} \frac{x^n}{n!}

This series converges for all real $x$ and is one of the most important series in mathematics. It provides a way to compute $e^x$ for any value of $x$.

Key Observation: For small values of $x$, we can often use just the first few terms of these series as approximations:

$\sin(x) \approx x$ (for small $x$)
$\cos(x) \approx 1 - \frac{x^2}{2}$ (for small $x$)
$e^x \approx 1 + x$ (for small $x$)

When only the first-order term is kept, the approximation is called a linear approximation.

Common Taylor Series Approximations

The following table provides three-term Taylor series approximations for common mathematical functions. These approximations are valid for small values of $x$ (or small values of the deviation from the expansion point). The table shows the first three non-zero terms in each expansion.

Function	Taylor Series Approximation	Valid For
$e^x$	$1 + x + \frac{x^2}{2} + \cdots$	$\|x\| \ll 1$
$e^{-x}$	$1 - x + \frac{x^2}{2} + \cdots$	$\|x\| \ll 1$
$\sin(x)$	$x - \frac{x^3}{6} + \cdots$	$\|x\| \ll 1$ (radians)
$\cos(x)$	$1 - \frac{x^2}{2} + \frac{x^4}{24} + \cdots$	$\|x\| \ll 1$ (radians)
$\tan(x)$	$x + \frac{x^3}{3} + \cdots $	$\|x\| \ll 1$ (radians)
$\ln(1+x)$	$x - \frac{x^2}{2} + \frac{x^3}{3} + \cdots$	$\|x\| \ll 1$
$(1+x)^n$	$1 + nx + \frac{n(n-1)}{2}x^2 + \cdots$	$\|x\| \ll 1$
$\sinh(x)$	$x + \frac{x^3}{6} + \cdots$	$\|x\| \ll 1$
$\cosh(x)$	$1 + \frac{x^2}{2} + \frac{x^4}{24} + \cdots$	$\|x\| \ll 1$
$\arcsin(x)$	$x + \frac{x^3}{6} + \cdots$	$\|x\| \ll 1$
$\arctan(x)$	$x - \frac{x^3}{3} + \cdots$	$\|x\| \ll 1$

Note: For trigonometric functions, $x$ must be in radians. The three-term approximations are more accurate than the two-term versions. For example, $\sin(x) \approx x - x^3/6$ is accurate to within 1% for $|x| < 0.5$ radians (about 29°), compared to $|x| < 0.24$ radians for the one-term approximation.

These approximations are particularly useful in physics when analyzing systems near equilibrium or when making small-angle approximations. They allow us to simplify complex expressions and gain physical insight into the behavior of systems.

Summary

Maclaurin Series: A Taylor series expansion centered at $x = 0$: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n$
Taylor Series: A more general expansion around $x = a$: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n$
Common Series: $\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$, $\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$, $e^x = 1 + x + \frac{x^2}{2!} + \cdots$
Applications: Taylor series are useful for approximating functions near a point, analyzing sensitivity to parameters, and simplifying complex expressions in physics problems.