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| Binomial Theorem for Fractional Index |
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| For any rational number n, |
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| We accept this expansion without proof. |
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| The restriction on x is not required when n is a natural number. |
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| Now, we shall see that when n is a natural number, then the above expansion coincides with that as given earlier. |
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Let n  N and |x|<1, then we have |
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| This is the same expansion as would have given by the binomial theorem for positive integral index. |
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| Some Observations: |
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 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. |
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 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. |
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 In the expansion of (1 + x)n, the exponent of x goes on increasing through 0. |
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 If n  N, then the coefficient of any term in (1 + x)n is nCk where k is the exponent of x. |
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 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. |
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Working rules for expanding (1 + x)n, n  Q: |
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Step 1: a) If n  N, then (1 + x)n can be expanded for all values of x and has (n+1) terms. |
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b) If n  Q - N, then (1 + x)n can be expanded only when |x|<1 and has infinitely many terms. |
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| Step 2: The first term in (1 + x)n is always 1. |
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| Step 3: The second terms is the product 'nx' of n and x. |
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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. |
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