The Taylor Series and Binomial Expansion

Suppose $f(x)$ is a continuous and infinitely differentiable function. Let $x = x_0 + \Delta x$ for some $\Delta x$ that is ``small''. Then the following is true:

$$f(x_0 + \Delta x) = f(x)\bigg\vert _{x = x_0} + \frac{d f}{d x}\bigg\vert _{x = x_0} \Delta x + \frac{1}{2!} \frac{d ^2f}{d x^2}\bigg\vert _{x = x_0} \Delta x^2$$

$$+ \frac{1}{3!} \frac{d ^3f}{d x^3}\bigg\vert _{x = x_0} \Delta x^3 + \ldots$$ (53)

This sum will always converge to the function value (for smooth functions and small enough $\Delta x$ ) if carried out to a high enough degree. Note well that the Taylor series can be rearranged to become the definition of the derivative of a function:

$$\frac{d f}{d x}\bigg \vert _{x = x_0} = \lim_{\Delta x \to 0} \frac{f(x_0 + \Delta x) - f(x_0)}{\Delta x} + {\cal O}(\Delta x)$$ (54)

where the latter symbols stands for ``terms of order $\Delta x$ or smaller'' and vanishes in the limit. It can similarly be rearranged to form formal definitions for the second or higher order derivatives of a function, which turns out to be very useful in computational mathematics and physics.

We will find many uses for the Taylor series as we learn physics, because we will frequently be interested in the value of a function ``near'' some known value, or in the limit of very large or very small arguments. Note well that the Taylor series expansion for any polynomial is that polynomial, possibly re-expressed around the new ``origin'' represented by $x_0$ .

To this end we will find it very convenient to define the following binomial expansion. Suppose we have a function that can be written in the form:

$$f(x) = (c + x)^n$$ (55)

where $n$ can be any real or complex number. We'd like expand this using the Taylor series in terms of a ``small'' parameter. We therefore factor out the larger of $x$ and $c$ from this expression. Suppose it is $c$ . Then:

$$f(x) = (c + x)^n = c^n(1 + \frac{x}{c})^n$$ (56)

where $x/c < 1$ . $x/c$ is now a suitable ``small parameter'' and we can expand this expression around $x = 0$ :

$$f(x) = c^n\left( 1 + n \frac{x}{c} + \frac{1}{2!}n(n-1)\left(\frac{x}{c}\right)^2 \right.$$

$$\left. \quad\quad\quad+ \frac{1}{3!}n(n-1)(n-2)\left(\frac{x}{c}\right)^3 + \ldots \right)$$ (57)

Evaluate the derivatives of a Taylor series around $x = 0$ to verify this expansion. Similarly, if $x$ were the larger we could factor out the $x$ and expand in powers of $c/x$ as our small parameter around $c = 0$ . In that case we'd get:

$$f(x) = x^n\left( 1 + n \frac{c}{x} + \frac{1}{2!}n(n-1)\left(\frac{c}{x}\right)^2 \right.$$

$$\left. \quad\quad\quad+ \frac{1}{3!}n(n-1)(n-2)\left(\frac{c}{x}\right)^3 + \ldots \right)$$ (58)

Remember, $n$ is arbitrary in this expression but you should also verify that if $n$ is any positive integer, the series terminates and you recover $(c + x)^n$ exactly. In this case the ``small'' requirement is no longer necessary.

We summarize both of these forms of the expansion by the part in the brackets. Let $y < 1$ and $n$ be an arbitrary real or complex number (although in this class we will use only $n$ real). Then:

$$(1 + y)^n = 1 + ny + \frac{1}{2!}n(n-1)y^2 + \frac{1}{3!}n(n-1)(n-2)y^3 + \ldots$$ (59)

This is the binomial expansion, and is very useful in physics.