# Binomial Theorem for Any Index

Let n be a rational number and x be a real number such that |x| < 1, then

xr + ... âˆž

Notes:
1. The condition |x| < 1 is unnecessary, if nis a whole number while the same condition is essential if n is a rational number other than a whole number.
2. Note that there are infinite number of term in the expansion of (1 + x)n, whenn is a negative integer or a fraction.
3. 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:

The expansion is valid when  < 1 or equivalently |x| < |a|.
4. Expansion of (x + a)n for any rational index

Case I When x > a, i.e.,

In this case,

Case II When x < a, i.e.,

In this case,

5. If n is a positive integer the above expansion contains (n + 1) terms and coincides with

(1 + x)n = nC0 + nC1x + nC2 x2 + ... + nCn xn, because nC0 = 1, nC1 = n

The general term in the expansion of (1 +x)n is given by
6. Let n is positive integer then by replacing n by â€“ n in the expansion for (1 + x)n, we get
= 1 â€“ nC1x + n+1C2x2 â€“ n+2C3x3 + ... +n+râ€“1Cr(â€“x)r + ...

Now replacing x by â€“x and n by â€“n in the expression of (1 + x)n, we get

= 1 + nC1x + n+1C2x2 + n+2C3x3 + ... + n+râ€“1Crxr + ...