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Summation problem

  1. Apr 27, 2015 #1
    1. The problem statement, all variables and given/known data
    If ##|P(x)|<=|e^{x-1}-1|## for all x> 0 where ##P(x)=\sum\limits_{r=0}^na_rx^r## then prove that ##|\sum\limits_{r=0}^nra_r|<=1##
    2. Relevant equations

    None
    3. The attempt at a solution

    ##P(1)=a_0+a_1+....##
    If the constants are positive, then ##P(1)<=|e^0-1|##
    So P(1)<=0
    so ##a_0+a_1+a_2+.... <=1##
    But how do I prove that ##0a_0+1.a_1+2.a_2+....<=1##
     
  2. jcsd
  3. Apr 27, 2015 #2

    Dick

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    Science Advisor
    Homework Helper

    Uh, ##P(1)=0##. Say why that's true. Then you want to show ##|P'(1)| \le 1##. Think about considering what the derivative of ##e^{x-1}-1## at ##x=1## might tell you about that.
     
  4. Apr 28, 2015 #3
    |P(1)|<=0. So the only possibility for P(1) is zero.
    P'(1)=a1+2a2+3a3+.....
    Derivative of ##e^{x-1}-1## at x=1 is 1. So P'(1) lies between 1 and -1.
     
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