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Math Help - Question on exponentials in inequalities

  1. #1
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    Question on exponentials in inequalities

    Using the fact that if \sum|a_n| converges then:
    |\sum_{n=0}^{\infty}a_n|\leq\sum_{n=0}^{\infty}|a_  n|,
    prove that for all z\in\bar{D}(0;1) (punctured disc centre 0 radius 1),
    (3-e)|z|\leq|e^z-1|\leq(e-1)|z|.

    I haven't even got the first part yet. Ive tried manipulating e (as \sum_{n=0}^{\infty}\frac{1}{n!}) but can't get the right answer.
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  2. #2
    Super Member girdav's Avatar
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    Re: Question on exponentials in inequalities

    For the second inequality |e^z-1|\leq \sum_{n=1}^{\infty}\frac{|z|^n}{n!}\leq \sum_{n=1}^{+\infty}\frac{|z|}{n!}=|z|(e-1).
    Last edited by girdav; December 29th 2011 at 10:54 AM.
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  3. #3
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    Re: Question on exponentials in inequalities

    Ok, I get that. How about the first bit?
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  4. #4
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    Re: Question on exponentials in inequalities

    Quote Originally Posted by Speed1991 View Post
    Ok, I get that. How about the first bit?
    You aren't asked to prove the first bit, only to use it...
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  5. #5
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    Re: Question on exponentials in inequalities

    Sorry, I meant the first part of the inequality.
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