# Binomial Theorem for Any Index

Let

*n*be a rational number and*x*be a real number such that |*x*| < 1, then*x*+ ... âˆž

^{r}

*Notes:*- The condition |
*x*| < 1 is unnecessary, if*n*is a whole number while the same condition is essential if*n*is a rational number other than a whole number. - Note that there are infinite number of term in the expansion of (1 +
*x*), when^{n}*n*is a negative integer or a fraction. - In the above expansion the first term is unity, If the first term is not unity and the index of the binomial is either a negative integer or a fraction, then we expand as follows:
*x*| < |*a*|. - Expansion of (
*x*+*a*)for any rational index^{n}**Case I**When*x*>*a*, i.e.,**Case II**When*x*<*a,*i.e., - If
*n*is a positive integer the above expansion contains (*n*+ 1) terms and coincides with*x*)=^{n}^{n}C_{0}+^{n}C_{1}*x*+^{n}C_{2}*x*^{2}+ ... +^{n}C_{n}*x*, because^{n}^{n}C_{0}= 1,^{n}C_{1}=*n*,*x*)is given by^{n} - Let
*n*is positive integer then by replacing*n*by â€“*n*in the expansion for (1 +*x*), we get^{n}

= 1 â€“^{n}C_{1}*x*+^{n}^{+1}*C*_{2}*x*^{2}â€“^{n}^{+2}*C*_{3}*x*^{3}+ ... +^{n}^{+râ€“1}*C*(â€“_{r}*x*)+ ...^{r}*x*by â€“*x*and*n*by â€“*n*in the expression of (1 +*x*), we get^{n}

= 1 +^{n}C_{1}*x*+^{n}^{+1}*C*_{2}*x*^{2}+^{n}^{+2}*C*_{3}*x*^{3}+ ... +^{n}^{+râ€“1}*C*+ ..._{r}x^{r}