Binomial Theorem for Rational Index

Binomial Theorem for Rational Index

1) If m is rational number and x is real such that |x| < 1 then \({{\left( 1+x \right)}^{m}}=1+\frac{m}{1!}x+\frac{m\left( m-1 \right)}{2!}{{x}^{2}}+…\).

\(=\,1+\sum\limits_{r=1}^{\infty }{\frac{m\left( m-1 \right)\left( m-r+1 \right)}{r!}}{{x}^{r}}\).

2) m = p/q [q ≠ 0] then p/q ϵ R⁺ excepting integers \({{\left( 1+x \right)}^{p/q}}=1+{{\left( \frac{p}{q} \right)}^{x}}+\left( \frac{p}{q} \right)\left( \frac{p}{q}-1 \right)\frac{{{x}^{2}}}{2!}\,+\,…\,+{{\frac{\left( \frac{p}{q} \right)\left( \frac{p}{q}-1 \right)……\left( \frac{p}{q}-r+1 \right)x}{r!}}^{r}}+\,…\),

\(=1+p\left( \frac{x}{q} \right)+\frac{p\left( p-q \right)}{2!}{{\left( \frac{x}{q} \right)}^{2}}+…+\frac{p\left( p-q \right)…\left( p-q\left( r-1 \right) \right)}{2!}{{\left( \frac{r}{q} \right)}^{2}}\),
\({{T}_{r+1}}=\frac{\left( \frac{p}{q} \right)\left( \frac{p}{q}-1 \right)……\left( \frac{p}{q}-r+1 \right)}{r!}{{x}^{r}}\) for r ≥ 1.

3) \({{\left( 1-x \right)}^{p/q}}=1-\frac{p}{1!}\left( \frac{x}{q} \right)+\frac{p\left( p-q \right)}{2!}{{\left( \frac{x}{q} \right)}^{2}}+\,…\,+\frac{{{\left( -1 \right)}^{r}}\left( p-q \right)…\left( p-q\left( r-1 \right) \right)}{r!}{{\left( \frac{x}{q} \right)}^{r}}….\).

4) \({{\left( 1-x \right)}^{-}}^{p/q}=1-\frac{p}{1!}\left( \frac{x}{q} \right)+\frac{p\left( p+q \right)}{2!}{{\left( \frac{x}{q} \right)}^{2}}+….+\frac{p\left( p+q \right)…\left( p+q\left( r-1 \right)q \right)}{r!}{{\left( \frac{x}{q} \right)}^{r}}….\).