Binomial Theorem
12.0 Binomial theorem for negative and fractional indices
12.0 Binomial theorem for negative and fractional indices
If $n \in R$, then $${(1 + x)^n} = 1 + nx + \frac{{n(n - 1)}}{{2!}}{x^2} + \frac{{n(n - 1)(n - 2)}}{{3!}}{x^3} + ... + \frac{{n(n - 1)(n - 2)...(n - r + 1)}}{{r!}}{x^r} + ...\infty $$
- The above expansion is valid for any rational number other than a whole number if $\left| x \right| < 1$.
- When the index is negative integer or a fraction then the number of terms in the expansion of ${(1 + x)^n}$ is infinite.
- The first term in the expansion must be unity when index is a negative integer or fraction. \[{(x + y)^n} = \left\{{\begin{array}{c}{{x^n}{{\left( {1 + \frac{y}{x}} \right)}^n} = {x^n}\left\{ {1 + n.\frac{y}{x} + \frac{{n(n - 1)}}{{2!}}{{\left( {\frac{y}{x}} \right)}^2} + ...} \right\}{\text{ if }}\left| {\frac{y}{x}} \right| < 1} \\ {{y^n}{{\left( {1 + \frac{x}{y}} \right)}^n} = {y^n}\left\{ {1 + n.\frac{x}{y} + \frac{{n(n - 1)}}{{2!}}{{\left( {\frac{x}{y}} \right)}^2} + ...} \right\}{\text{ if }}\left| {\frac{x}{y}} \right| < 1} \end{array}} \right.\]
- The general term in the expansion of ${(1 + x)^n}$ is $${T_{r + 1}} = \frac{{n(n - 1)(n - 2)...(n - r + 1)}}{{r!}}{x^r}$$
Expansions to be remember $(\left| x \right| < 1)$
1. ${(1 + x)^{ - 1}} = 1 - x + {x^2} - {x^3} + ... + {( - 1)^r}{x^r} + ...\infty $
2. ${(1 - x)^{ - 1}} = 1 + x + {x^2} + {x^3} + ... + {x^r} + ...\infty $
3. ${(1 + x)^{ - 2}} = 1 - 2x + 3{x^2} - 4{x^3} + ... + {( - 1)^r}(r + 1){x^r} + ...\infty $
4. ${(1 - x)^{ - 2}} = 1 + 2x + 3{x^2} + 4{x^3} + ... + (r + 1){x^r} + ...\infty $
