For the binomial \((1+x)^n\), where \(n\) is any
rational number, e.g. fractions and negative numbers, it can be expanded according to this more useful formula, for the first 4 terms:
\((1+x)^n = 1 + nx + \dfrac{n(n-1)}{2!} x^2 + \dfrac{n(n-1)(n-2)}{3!} x^3 + ... \)
Partial fractions are sometimes used to generate expressions for binomial expansion.
Validity of binomial expansion
For \((p+qx)^n\), when \(n\) is any rational number, the above binomial expansion formula is only
valid when \(\Big|\dfrac{qx}{p}\Big|<1\).
\((1+x)^n = 1 + nx + \dfrac{n(n-1)}{2!} x^2 + \dfrac{n(n-1)(n-2)}{3!} x^3 + ... \)
For \((p+qx)^n\), when \(n\) is any rational number, the above binomial expansion formula is only valid when \(\Big|\dfrac{qx}{p}\Big|<1\).
3