My Math Forum  

Go Back   My Math Forum > High School Math Forum > Algebra

Algebra Pre-Algebra and Basic Algebra Math Forum


Thanks Tree2Thanks
  • 1 Post By tahirimanov19
  • 1 Post By tahirimanov19
Reply
 
LinkBack Thread Tools Display Modes
October 9th, 2019, 08:07 AM   #1
Senior Member
 
Joined: Dec 2015
From: somewhere

Posts: 734
Thanks: 98

Inequalities #3

Prove that $\displaystyle \: \displaystyle (1+e^{-1})\cdot (1+e^{-2}) \cdot ... \cdot (1+e^{-n}) < \displaystyle (e^{-1^2 }+e^{-2^2 }+e^{-3^2}+...+e^{-n^2 } )^{\displaystyle -1} \; ,n >1$ .
e-euler constant . Method required !

Last edited by idontknow; October 9th, 2019 at 08:14 AM.
idontknow is offline  
 
October 9th, 2019, 09:07 AM   #2
Senior Member
 
Joined: Mar 2015
From: Universe 2.71828i3.14159

Posts: 132
Thanks: 49

Math Focus: Area of Circle
Using Bernoulli inequality:

$L>1+e^{-1} + e^{-2}+...+e^{-n}$

Show that, $e^{-n}>e^{-n^2}$. Therefore,

$L>1+e^{-1} + e^{-2}+...+e^{-n}>e^{-1^2}+e^{-2^2}+...+e^{-n^2}$

Invert the previous inequality, you get,

$L<(e^{-1^2}+e^{-2^2}+...+e^{-n^2})^{-1}$

L stands for the left side of inequality.
Thanks from idontknow

Last edited by tahirimanov19; October 9th, 2019 at 09:12 AM.
tahirimanov19 is offline  
October 9th, 2019, 09:10 AM   #3
Senior Member
 
Joined: Mar 2015
From: Universe 2.71828i3.14159

Posts: 132
Thanks: 49

Math Focus: Area of Circle
$(1+x_1)(1+x_2)+...+(1+x_n) \ge 1+x_1+x_2+...+x_n$

$x_1,x_2,...,x_n>-1$ and $ sgn(x_1)=sgn(x_2)=...=sgn(x_n)$

------

Also, for any x>-1,

$(1+x)^n \ge 1+nx, \; n \in \mathbb{N}, n>1$.

And equality is true iff x=0.

------

Prove these.
tahirimanov19 is offline  
October 9th, 2019, 09:24 AM   #4
Senior Member
 
Joined: Mar 2015
From: Universe 2.71828i3.14159

Posts: 132
Thanks: 49

Math Focus: Area of Circle
If you want inequalities, I can post plenty of them...
Thanks from idontknow
tahirimanov19 is offline  
Reply

  My Math Forum > High School Math Forum > Algebra

Tags
inequalities



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
how to selecte 2 inequalities between three inequalities for solving linear programmi mj1395 Elementary Math 2 July 18th, 2016 08:10 PM
inequalities Alexis87 Algebra 3 November 20th, 2013 04:57 AM
Help with inequalities drewm Algebra 1 July 2nd, 2011 07:10 PM
inequalities outsos Real Analysis 15 January 1st, 2011 05:38 PM
Inequalities TungLHang Algebra 17 November 13th, 2010 04:44 AM





Copyright © 2019 My Math Forum. All rights reserved.