# Summation problem

Tags:
1. Apr 27, 2015

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. Apr 27, 2015

### Dick

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. Apr 28, 2015

Derivative of $e^{x-1}-1$ at x=1 is 1. So P'(1) lies between 1 and -1.