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Thread: 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 $\displaystyle \sqrt{1+x}$?
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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 $\displaystyle \sqrt{1+x}$?
    $\displaystyle \sqrt{1+x}$ can be written as $\displaystyle (1+x)^{\frac{1}{2}}$.

    You just use the usual binomial formula but with the value $\displaystyle \frac{1}{2}$
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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 $\displaystyle \sqrt{1+x}$?
    If you have $\displaystyle \sqrt{1+x}$ as your exponent, e.g., $\displaystyle (a+b)^{\sqrt{1+x}}$, 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
    $\displaystyle \sqrt{1+x}$ can be written as $\displaystyle (1+x)^{\frac{1}{2}}$.

    You just use the usual binomial formula but with the value $\displaystyle \frac{1}{2}$
    Then how do you calculate the factorial for $\displaystyle \sum_{k=0}^{n} {\frac{1}{2} \choose k}x^k$? 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 $\displaystyle \frac{(r)_k}{k!}$ meant initially but got it. Thanks again.
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