Summation problem

  • #1
528
33

Homework Statement


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##

Homework Equations



None

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##
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,260
619

Homework Statement


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##

Homework Equations



None

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##

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.
 
  • #3
528
33
|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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