This is one particular power series that we can use to expand the expression (1+x)n, for any value of n. If n is a positive whole number, then we could expand it quite well anyway, by multiplying out the brackets (although if n is large this could take some time!).
But the good thing is that the binomial expansion works if n is negative as well, or even if n is a fraction. The binomial series gives us a formula that we can use, whatever n is.
Here are the first few terms of the formula for (1+x)n:
What this means is, that for any value of n, we can rewrite (1+x)n as this series of powers of x.
We've got just enough terms there for the pattern to start to show. As you can see, the terms form a power series, with the power of x increasing by one every time. The expressions in front of the powers of x are the coefficients.
There is a definite pattern to those coefficients. For a start look at the numbers on the top in each case. First n, then n(n-1), then n(n-1)(n-2). Every time the power of x increases by 1, another set of brackets gets put on, and the number following the minus sign in those new brackets is one less than the power. So for the x3 term, the last brackets have (n-2) inside.
The other part of the pattern is the number on the bottom. This number is the same as the power of x, and is followed by an exclamation mark each time! That exclamation mark is the factorial function.
So now we know the pattern, we could write down the coefficient for any power of x in the series. Consider the term x8. What would be the coefficient in that case?
We can now write down the expansion of (1+x)n, whatever the value of n. Try using it to write down the expansion of (1+x)6, for example. You should find that the series goes up to x6, then stops.
The reason the series was finite in that case, i.e. it stopped after a certain number of terms, was that n was a positive whole number. That meant that sooner or later we got a bracket (n-6) on the top, which was zero, since n=6. All terms after that would have that (n-6) in, so all would be zero.
However, if n was negative, say n=-3, then we'd never get a zero on the top, since that would need (n+3), which we never get. Similarly if n is a fraction we never get zero on top, since the numbers in the brackets are always whole numbers. So in the cases when n is negative or fractional, the binomial series is an infinite series.
Here's an example with n negative: write down the first 4 terms of the series for (1+x)-4.
Here's an example with n a fraction: write down the first 4 terms of the series for (1+x)3/2.