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Math Help - Why can this expression be approximated by that other expression?

  1. #1
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    Why can this expression be approximated by that other expression?

    This expression:

    \frac{a^m}{(1+a)^m}

    can be approximated by this expression for a >> 1 (i guess a is ten or more)

    1-\frac{m}{a}

    Why?
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  2. #2
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    Re: Why can this expression be approximated by that other expression?

    I might be inclined to go something along these lines...

    $\displaystyle \begin{align*} \frac{a^m}{ \left( 1 + a \right) ^m } &= \left( \frac{a}{1 + a} \right) ^m \\ &= \left( \frac{1 + a - 1 }{1 + a} \right) ^m \\ &= \left( 1 - \frac{1}{1 + a} \right) ^m \\ &= \left[ 1 - \frac{1}{1 - \left( - a \right) } \right] ^m \\ &= \left\{ 1 - \sum_{n = 0}^{\infty}{ \left[ \left( -a \right) ^n \right] } \right\} ^m \\ &= \left\{ - \sum_{n = 1}^{\infty} \left[ \left( -a \right) ^n \right] \right\} ^m \end{align*}$

    And now you can probably get a decent approximation using a binomial expansion. Of course this is only convergent where |a| < 1.
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  3. #3
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    Re: Why can this expression be approximated by that other expression?

    Thanks a bunch, Prove It! You sorted this out for me perfectly.
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