For any real number \(n\),

This expansion is valid for \(|x|<1\) when \(n\) is negative or fractional.

\[ (1+x)^n = 1 + nx + \frac{n(n-1)}{2}x^2 + \frac{n(n-1)(n-2)}{6}x^3 + \cdots \]

In many questions, you are asked only for the first 2, 3, or 4 terms.

If \(n\) is a positive integer, the expansion stops after a finite number of terms.

Example:

\[ (1+x)^3 = 1 + 3x + 3x^2 + x^3 \]

This is the usual binomial expansion from earlier algebra.

If \(n\) is negative, the expansion does not end. It becomes an infinite series.

Example: Expand \((1+x)^{-1}\)

\[ (1+x)^{-1} = 1 + (-1)x + \frac{(-1)(-2)}{2}x^2 + \frac{(-1)(-2)(-3)}{6}x^3 + \cdots \]

Notice that the signs alternate.

Expand \((1+2x)^{-2}\) up to the term in \(x^3\).

Using

\[ (1+u)^n = 1 + nu + \frac{n(n-1)}{2}u^2 + \frac{n(n-1)(n-2)}{6}u^3 + \cdots \]

with \(n=-2\) and \(u=2x\):

\[ (1+2x)^{-2} = 1 + (-2)(2x) + \frac{(-2)(-3)}{2}(2x)^2 + \frac{(-2)(-3)(-4)}{6}(2x)^3 + \cdots \]

\[ = 1 - 4x + 3(4x^2) - 4(8x^3) + \cdots \]

The same formula works for fractional powers.

Example: Expand \((1+x)^{1/2}\) up to the term in \(x^3\).

\[ (1+x)^{1/2} = 1 + \frac12 x + \frac{\frac12(-\frac12)}{2}x^2 + \frac{\frac12(-\frac12)(-\frac32)}{6}x^3 + \cdots \]

Often you must first rewrite the expression into the form \((1+u)^n\).

Example: Expand \((2-x)^{-1}\) up to the term in \(x^2\).

First factor out 2:

\[ (2-x)^{-1} = \left[2\left(1-\frac{x}{2}\right)\right]^{-1} \]

\[ = \frac12\left(1-\frac{x}{2}\right)^{-1} \]

Now use the expansion for \((1+u)^{-1}\):

\[ \left(1-\frac{x}{2}\right)^{-1} = 1 + \frac{x}{2} + \frac{x^2}{4} + \cdots \]

The general term in the expansion of \((1+x)^n\) is:

This is useful when finding a specific term.

Binomial expansions are often used for approximation when \(x\) is small.

Example:

\[ (1+x)^{-1} \approx 1-x \]

when \(|x|\) is very small.

• For negative powers, forgetting that the expansion is infinite.
• Not rewriting the expression into the form \((1+u)^n\) first.
• Sign errors when \(n\) is negative.
• Not substituting the whole \(u\) correctly, for example forgetting to square or cube it.
• Using the expansion outside its valid range \(|x|<1\) for non-integer powers.