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Math Help - binomial theorem for all powers

  1. #1
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    binomial theorem for all powers

    My book mentioned that
    "If -1<x<1, then (1+x)^n = 1 + nx + n(n-1)(x^2)/2! + n(n-1)(n-2)(x^3)/3! + ... + n(n-1)(n-2)...(n-r+1)(x^r)/r! + ..."
    Why -1<x<1? Why is this restriction on x necessary?
    Thanks in advance.
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  2. #2
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    Re: binomial theorem for all powers

    Quote Originally Posted by startanewww View Post
    My book mentioned that
    "If -1<x<1, then (1+x)^n = 1 + nx + n(n-1)(x^2)/2! + n(n-1)(n-2)(x^3)/3! + ... + n(n-1)(n-2)...(n-r+1)(x^r)/r! + ..."
    Why -1<x<1? Why is this restriction on x necessary?
    Thanks in advance.
    Because it's possible that infinite series might diverge (go to infinity). You need to check the interval for which the series is convergent.
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    Re: binomial theorem for all powers

    In that case, shouldn't the restriction be "x =/= +infinity or -infinity (when n is just a small number)"?
    If -1<x<1, will the value of (1+x)^n be very small?
    I really don't understand..
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  4. #4
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    Re: binomial theorem for all powers

    The series depends on x. But the series itself is only convergent if it follows specific conditions, which means that it will only be convergent for particular values of x (the ones which will satisfy the conditions). I suggest you research convergence tests for infinite series.
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    Re: binomial theorem for all powers

    Quote Originally Posted by startanewww View Post
    My book mentioned that
    "If -1<x<1, then (1+x)^n = 1 + nx + n(n-1)(x^2)/2! + n(n-1)(n-2)(x^3)/3! + ... + n(n-1)(n-2)...(n-r+1)(x^r)/r! + ..."
    Why -1<x<1? Why is this restriction on x necessary?
    I think that you are missing the point.
    {\left( {1 + x} \right)^n} = \sum\limits_{k = 0}^n {{\binom{n}{k} x^k}} for any x\in\mathbb{R}.
    So it would be true for -1<x<1.
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    Re: binomial theorem for all powers

    Quote Originally Posted by Plato View Post
    I think that you are missing the point.
    {\left( {1 + x} \right)^n} = \sum\limits_{k = 0}^n {{\binom{n}{k} x^k}} for any x\in\mathbb{R}.
    So it would be true for -1<x<1.
    The OP was actually referring to the GENERALISED Binomial Theorem, which is an infinite series.
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