AdityaDev said: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##
The summation problem is a mathematical concept that involves finding the sum of a sequence of numbers. It is typically denoted by the symbol Σ and is commonly encountered in calculus and other areas of mathematics.
P(x) is a function that represents the sum of a sequence of numbers. It is often used in conjunction with the summation problem to find the value of the sequence at a specific point or to determine the limit of the sequence.
The limit of |e^(x-1)-1| for x>0 is equal to 0. This means that as x approaches positive infinity, the value of the expression e^(x-1)-1 gets closer and closer to 0.
The summation problem and the limit of |e^(x-1)-1| for x>0 are closely related because the limit of the expression is often used to determine the value of the summation problem at a specific point. Additionally, the limit can also be used to determine whether a series of numbers converges or diverges.
Proving the summation problem and its limit is important because it allows us to understand the behavior of sequences of numbers and to accurately calculate their values. It also has many practical applications in fields such as physics, engineering, and economics, where the use of series and summations is common.