1.4.8. Taylor Series and Approximations

Table of Contents

Why Approximation Matters in Physics

Physics often uses formulas that are exact, but exact formulas can be hard to work with. A complicated function may describe nature perfectly, yet in many situations we only need to know what happens near one particular value. Taylor series give a systematic way to replace a difficult function by a simpler polynomial that behaves almost the same in a small region.

This is extremely useful because polynomials are easy to add, differentiate, integrate, and solve approximately. In physics, many important results come from keeping only the first few terms of a Taylor series.

The Basic Idea of a Taylor Series

Suppose a function $f(x)$ is smooth enough near some point $x=a$. Then we can write it as

$$
f(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+\cdots
$$

This is called the Taylor series of $f(x)$ about the point $x=a$.

Each term uses information about the function at the single point $a$, namely its value and its derivatives there. The closer $x$ is to $a$, the more accurate the approximation usually is when only a few terms are kept.

If the expansion is around $a=0$, the series is called a Maclaurin series:

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

Important formula:
$$
f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n
$$
Maclaurin series is the special case $a=0$:
$$
f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n
$$

Approximations from Truncating the Series

In practice, we usually do not keep infinitely many terms. We stop after one, two, or three terms. This is called truncating the series.

For example,

$$
f(x)\approx f(a)+f'(a)(x-a)
$$

is the linear approximation.

Adding one more term gives the quadratic approximation:

$$
f(x)\approx f(a)+f'(a)(x-a)+\frac{f''(a)}{2}(x-a)^2
$$

These approximations are most useful when $x-a$ is small.

The meaning of "small" depends on the problem. If $(x-a)^2$ is much smaller than $(x-a)$, then the quadratic and higher terms may be negligible compared with the linear term.

A Taylor approximation is reliable when the quantity $(x-a)$ is small enough that higher powers,
$$
(x-a)^2,\ (x-a)^3,\ (x-a)^4,\dots
$$
become progressively less important.

Common Maclaurin Series

Some functions appear constantly in physics. Their series are worth remembering.

Function	Maclaurin series
$e^x$	$1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\cdots$
$\sin x$	$x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\cdots$
$\cos x$	$1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\cdots$
$\ln(1+x)$	$x-\dfrac{x^2}{2}+\dfrac{x^3}{3}-\cdots$
$(1+x)^n$	$1+nx+\dfrac{n(n-1)}{2!}x^2+\cdots$

For small $x$, these immediately give useful approximations:

$$
e^x \approx 1+x
$$

$$
\sin x \approx x
$$

$$
\cos x \approx 1-\frac{x^2}{2}
$$

$$
\ln(1+x)\approx x
$$

$$
(1+x)^n \approx 1+nx
$$

Small Angle Approximations

One of the most important uses in beginner physics is the small angle approximation. When an angle $\theta$ is measured in radians and is small, the trigonometric functions simplify:

$$
\sin\theta \approx \theta
$$

$$
\tan\theta \approx \theta
$$

$$
\cos\theta \approx 1-\frac{\theta^2}{2}
$$

These come directly from the Maclaurin series of sine and cosine.

This is why many oscillation formulas become simple for small displacements. The approximation works well only when $\theta$ is small and measured in radians, not degrees.

Small angle approximations:
$$
\sin\theta \approx \theta,\qquad \tan\theta \approx \theta,\qquad \cos\theta \approx 1-\frac{\theta^2}{2}
$$
These are valid for small $\theta$ in radians.

Visual Meaning

A Taylor approximation matches the graph of a function near the expansion point. The linear approximation matches the slope there. The quadratic approximation also captures how the curve bends.

Rendering scientific sketch…

Function and linear approximation near a point

Near $x=a$, the dashed line is close to the true curve. Farther away, the difference grows.

Example, Approximating a Function Near a Point

Consider

$$
f(x)=\sqrt{1+x}
$$

We expand around $x=0$. Using the binomial series with $n=\frac{1}{2}$,

$$
(1+x)^{1/2}=1+\frac{1}{2}x+\frac{\frac{1}{2}\left(\frac{1}{2}-1\right)}{2!}x^2+\cdots
$$

$$
\sqrt{1+x}=1+\frac{x}{2}-\frac{x^2}{8}+\cdots
$$

If $x$ is small, then

$$
\sqrt{1+x}\approx 1+\frac{x}{2}
$$

For instance, if $x=0.04$,

$$
\sqrt{1.04}\approx 1+\frac{0.04}{2}=1.02
$$

The exact value is about $1.0199$, so the approximation is very good.

Example, Exponential Approximation

Suppose a formula contains $e^{\varepsilon}$ where $\varepsilon$ is very small. Then

$$
e^{\varepsilon}=1+\varepsilon+\frac{\varepsilon^2}{2}+\cdots
$$

If $\varepsilon$ is tiny, we often use

$$
e^{\varepsilon}\approx 1+\varepsilon
$$

This kind of approximation appears often when studying growth, decay, oscillations, and wave behavior.

Example, Small Oscillation Style Approximation

Consider the expression

$$
1-\cos\theta
$$

Using

$$
\cos\theta \approx 1-\frac{\theta^2}{2}
$$

we get

$$
1-\cos\theta \approx \frac{\theta^2}{2}
$$

This result appears frequently in geometry and mechanics when angles are small.

How to Build a Taylor Approximation

To approximate a function near $x=a$, the steps are simple. First choose the point $a$ where the function is to be expanded. Next compute the derivatives of the function. Then evaluate those derivatives at $x=a$. Finally substitute them into the Taylor formula and keep as many terms as needed.

The first few terms often already give an excellent result. The best choice of $a$ is usually the point near which you want to study the function.

Order of Approximation

Physicists often describe approximations by order. For example,

$$
f(x)=1+x+x^2+\cdots
$$

If we keep only

$$
f(x)\approx 1+x
$$

we call this a first order approximation in $x$. If we keep

$$
f(x)\approx 1+x+x^2
$$

it is a second order approximation.

This language tells us how many powers of the small quantity have been retained.

If a quantity $\varepsilon$ is small, then
first order means keeping terms up to $\varepsilon$,
second order means keeping terms up to $\varepsilon^2$,
and neglecting higher powers such as $\varepsilon^3$, $\varepsilon^4$, and so on.

Error and Validity

A Taylor approximation is not exact unless all terms are included. The neglected terms create the error. Usually, the size of the first omitted term gives a rough idea of the accuracy.

For example, if

$$
\sin x \approx x
$$

then the next term is

$$
-\frac{x^3}{6}
$$

So the error is expected to be of order $x^3$ for small $x$.

This helps us judge whether an approximation is acceptable. If $x=0.1$, then $x^3=0.001$, which is much smaller than $x=0.1$. But if $x=1$, the neglected term is no longer tiny, and the approximation becomes much worse.

Why This Is Powerful in Physics

Many physical laws become easier when changes are small. Near equilibrium, near zero angle, near a reference value, or for weak disturbances, Taylor series turn difficult functions into simple algebraic forms. This is one of the main tools for creating workable physical models.

A complicated exact expression can often be replaced by a low order polynomial that captures the essential behavior in the region of interest. That is why Taylor series are used throughout mechanics, waves, thermodynamics, electromagnetism, optics, and modern physics.

Summary

Taylor series express a smooth function as a sum of powers of $(x-a)$ built from derivatives at $a$. Truncating the series gives approximations that are especially useful when $(x-a)$ is small. Some of the most important approximations in physics come from the series for exponential, trigonometric, logarithmic, and power functions. Small angle approximations are a major example. The key idea is always the same, replace a difficult function by a simpler one that is accurate in a limited region.

Core idea:
For a small quantity $\varepsilon$, a complicated function can often be simplified by keeping only the first few terms of its Taylor series.
Examples:
$$
e^\varepsilon \approx 1+\varepsilon
$$
$$
\sin\varepsilon \approx \varepsilon
$$
$$
\cos\varepsilon \approx 1-\frac{\varepsilon^2}{2}
$$
$$
(1+\varepsilon)^n \approx 1+n\varepsilon
$$

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