The Binomial Series

The binomial series is simply the expansion of $(1 + x)^k$ into powers of $x$, for any exponent $k$ at all. ("Binomial" means "two terms", here the $1$ and the $x$ being raised to a power.) It's good for turning awkward things like square roots and $\frac{1}{\sqrt{1 - v^2/c^2}}$ into simple polynomials you can estimate by hand. The coefficient $\binom{k}{n} = \frac{k(k-1)\cdots(k-n+1)}{n!}$ looks fussy, but it's just $k$ times one less, times one less again, $n$ factors in all, divided by $n!$. It's the same recipe as Pascal's triangle, so bear with it.

Where does it come from? It's just the Taylor series of $(1 + x)^k$ at $0$. Each derivative brings down the exponent and lowers it by one: $k$, then $k(k-1)$, then $k(k-1)(k-2)$, and dividing by $n!$ gives $\binom{k}{n}$.

When $k$ is a positive whole number, one of those factors eventually hits $k - k = 0$, so every coefficient after $n = k$ is zero and you get the ordinary binomial theorem. For any other $k$ no factor is ever zero, so the series never stops, and it converges for $|x| < 1$.

The first terms are worth memorizing:

Common cases:

• $\sqrt{1 + x} = 1 + \dfrac{x}{2} - \dfrac{x^2}{8} + \dfrac{x^3}{16} - \cdots$ ($k = \frac12$)
• $\dfrac{1}{1 + x} = 1 - x + x^2 - \cdots$ ($k = -1$, the geometric series again)
• $\dfrac{1}{\sqrt{1 - x^2}} = 1 + \dfrac{x^2}{2} + \dfrac{3x^4}{8} + \cdots$, which integrates to the series for $\arcsin x$.

A common mistake. It's tempting to use the formula on $\sqrt{4 + x}$ directly, as if the $4$ could stand in for the $1$. It can't: the series is only for $(1 + \text{small})^k$. So factor first, pulling out whatever makes the leading term $1$: $\sqrt{4 + x} = 2\left(1 + \frac{x}{4}\right)^{1/2}$. Now the small quantity is $\frac x4$, so the series is valid for $|x| < 4$.

In physics. The relativistic factor $\gamma = \left(1 - \frac{v^2}{c^2}\right)^{-1/2} \approx 1 + \frac{v^2}{2c^2}$, so kinetic energy $(\gamma - 1)mc^2 \approx \frac12 mv^2$. Notice what that says: Newton's formula is the first term of Einstein's. At everyday speeds $\frac{v^2}{c^2}$ is so tiny that the rest of the series doesn't matter, which is why Newton's physics worked for centuries.

Einstein's kinetic energy reduces to Newton's $\frac12 mv^2$ through the binomial series, and engineers use $(1 + x)^k \approx 1 + kx$ for quick error estimates: a 2% error in a radius gives about a 4% error in an area.