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Math Help - The binomial theorem when n is not an integer

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    The binomial theorem when n is not an integer

    How do we expand binomial expressions when n is not an integer, for example \sqrt{1+x}?
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    Super Member craig's Avatar
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    Quote Originally Posted by Hibachi View Post
    How do we expand binomial expressions when n is not an integer, for example \sqrt{1+x}?
    \sqrt{1+x} can be written as (1+x)^{\frac{1}{2}}.

    You just use the usual binomial formula but with the value \frac{1}{2}
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    MHF Contributor undefined's Avatar
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    Quote Originally Posted by Hibachi View Post
    How do we expand binomial expressions when n is not an integer, for example \sqrt{1+x}?
    If you have \sqrt{1+x} as your exponent, e.g., (a+b)^{\sqrt{1+x}}, then you won't be using binomial expansion. Perhaps you're supposed to use logarithms.

    Edit: Seems I misinterpreted your question. At any rate, maybe this link on Newton's generalized binomial theorem will help.
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    Quote Originally Posted by craig View Post
    \sqrt{1+x} can be written as (1+x)^{\frac{1}{2}}.

    You just use the usual binomial formula but with the value \frac{1}{2}
    Then how do you calculate the factorial for \sum_{k=0}^{n} {\frac{1}{2} \choose k}x^k? I thought they were defined for just (positive) integers. Also, the number of terms is not even finite, so I'm not sure whether I can use that formula.
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    Quote Originally Posted by undefined View Post
    Edit: Seems I misinterpreted your question.
    Sorry. It was my fault. I should have worded it better.
    At any rate, maybe this link on Newton's generalized binomial theorem will help.
    Got it! Thank you. Little bit got confused about as to what \frac{(r)_k}{k!} meant initially but got it. Thanks again.
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