Binomial Theorem for Fractional Index

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For any rational number n,

We accept this expansion without proof.

The restriction on x is not required when n is a natural number.

Now, we shall see that when n is a natural number, then the above expansion coincides with that as given earlier.

Let n N and |x|<1, then we have

This is the same expansion as would have given by the binomial theorem for positive integral index.

Some Observations:

• If n N, then (1 + x)ⁿ is defined for all values of x and if n Q - N, then (1 + x)ⁿ is defined only when |x|<1.

• If n N, then (1 + x)ⁿ contains only n+1 terms and if n Q - N, then (1 + x)ⁿ contains infinitely many terms.

• In the expansion of (1 + x)ⁿ, the exponent of x goes on increasing through 0.

• If n N, then the coefficient of any term in (1 + x)ⁿ is ⁿC_k where k is the exponent of x.

• If n N, then the exact value of (1 + x)ⁿ can be found by adding all terms (equal to n+1) in the expansion of (1 + x)ⁿ and if n Q - N, then only an approximate value of (1 + x)ⁿ can be found by adding certain finite number of terms in the expansion of (1 + x)ⁿ.

Working rules for expanding (1 + x)ⁿ

nQ:

Step 1: a)

If n N, then (1 + x)ⁿ can be expanded for all values of x and has (n+1) terms.

b) If n Q - N, then (1 + x)ⁿ can be expanded only when |x|<1 and has infinitely many terms.

Step 2:

The first term in (1 + x)ⁿ is always 1.

Step 3:

The second terms is the product 'nx' of n and x.

Step 4:

For the third term, take coefficient as increase the power of x by 1. Thus, the third term is Repeat this process repeatedly.