# Summation problem

## 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##

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##

Dick
Homework Helper

## 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##

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.