How to Prove the Inequality e^x > (1 +f(x)/n)^n for x in (0, infinity)?

  • Thread starter Thread starter regularngon
  • Start date Start date
  • Tags Tags
    Inequality
Click For Summary

Homework Help Overview

The discussion revolves around proving the inequality e^x > (1 + f(x)/n)^n for x in the interval (0, infinity), given that 0 <= f(x) < infinity. Participants are exploring the implications of this inequality within the context of mathematical analysis and series representations.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss using the series representation of e^x and the binomial theorem to analyze the inequality. Questions are raised regarding the conditions under which the inequality holds, particularly concerning the variable n and specific values of f(x).

Discussion Status

The discussion is ongoing, with participants questioning the validity of the inequality for various values of n and specific functions f(x). Some suggest that there may be a typographical error in the original statement, while others propose examining series expansions to clarify the relationship between the two sides of the inequality.

Contextual Notes

There is uncertainty regarding the conditions of the inequality, particularly whether it is meant to hold for all n or under specific limits. Additionally, the implications of particular values of f(x) are being scrutinized, indicating potential constraints on the problem.

regularngon
Messages
19
Reaction score
0

Homework Statement


If 0 <= f(x) < infinity, then I need to show that e^x > (1 +f(x)/n)^n for x in (0, infinity)


Homework Equations





The Attempt at a Solution


I'm pretty sure the answer lies in the comparison of the series representation for e^x and writing (1 +f(x)/n)^n out with the binomial theorem. I did so, however I still don't see it.
 
Physics news on Phys.org
Is this supposed to be true for all n, or the limit as approaches infinity or what? Suppose f(0)=1 then the statement is false for n=1, and x=0.
 
for all n, and x > 0 though.
 
regularngon said:
for all n, and x > 0 though.

Again take n=1, and any function f(x) such that f(1)=2, and it is false.
 
Yea it must be a typo on my teachers part. I'm going to guess he meant e^f(x).
 
Well in that case writing out the series expansions for both sides of the equation would help.
 

Similar threads

Prove that the limit as n->infinity of n^n/n! is infinity
  • · Replies 7 ·
Replies
7
Views
2K
Prove that ##\lim\limits_{x \to \infty} f(x) = 0##
  • · Replies 3 ·
Replies
3
Views
2K
Integration problem using inscribed rectangles
  • · Replies 14 ·
Replies
14
Views
2K
Finding the Maximum Value of f(x)=x(1-x)^n in [0,1]: A Calculus Problem
  • · Replies 3 ·
Replies
3
Views
2K
Proving that there exists a solution to ##f(x+\frac1n)-f(x)=0##
  • · Replies 4 ·
Replies
4
Views
1K
Proof given ##x < y < z## and a twice differentiable function
  • · Replies 8 ·
Replies
8
Views
2K
Proving limits for roots and exponents
  • · Replies 13 ·
Replies
13
Views
2K
Integral of x^n using Reimann sums
  • · Replies 4 ·
Replies
4
Views
2K
Problem 1-23 and 1-24 from Spivak's Calculus on Manifolds
  • · Replies 6 ·
Replies
6
Views
2K
How to show that these sequences are summable?
  • · Replies 1 ·
Replies
1
Views
2K