My Math Forum  

Go Back   My Math Forum > College Math Forum > Real Analysis

Real Analysis Real Analysis Math Forum


Thanks Tree2Thanks
  • 1 Post By mathman
  • 1 Post By The Epsilon
Reply
 
LinkBack Thread Tools Display Modes
January 5th, 2017, 06:42 AM   #1
Member
 
Joined: Oct 2012

Posts: 54
Thanks: 0

Show that

Show that : 1+1/2+1/3+....+1/n>2n/(n+1) for n>1
fahad nasir is offline  
 
January 5th, 2017, 02:30 PM   #2
Global Moderator
 
Joined: May 2007

Posts: 6,136
Thanks: 468

mathematical induction works.

1+1/2+...+1/n+1/(n+1)>2n/(n+1)+1/(n+1)=(2n+1)/(n+1)

Need to show: (2n+1)/(n+1)>2(n+1)/(n+2)
or $\displaystyle (2n+1)(n+2)>2(n+1)^2$ or $\displaystyle 2n^2+5n+2>2n^2+4n+2$.
Thanks from fahad nasir
mathman is offline  
January 5th, 2017, 04:16 PM   #3
Member
 
Joined: Dec 2016
From: USA

Posts: 43
Thanks: 9

It's a routine induction proof.

Verify the base case, $n = 2$.

Assume it's true for some positive integer $n > 1$.

Then, using that assumption, show that the claim holds for the "$n+1$" case.

Hint: Start by adding $1/(n+1)$ to both sides.
quasi is offline  
January 16th, 2017, 10:15 PM   #4
Newbie
 
Joined: Jan 2017
From: VN

Posts: 2
Thanks: 1

Quote:
Originally Posted by fahad nasir View Post
Show that : 1+1/2+1/3+....+1/n>2n/(n+1) for n>1
I use Gauss's idea that he use when he solves the problem '1+2+...+n+...+100'.

A useful inequality: with $a, b>0$, we obtain
$$ \frac{1}{a}+\frac{1}{b} \ge \frac{4}{a+b}.$$

Therefore,
$$\frac{1}{k}+\frac{1}{n+1-k} \ge \frac{4}{n+1}\,\forall k=1, 2, ..., n.$$
Hence,
$2\sum_{k=1}^{n}\frac{1}{k} \ge \frac{4n}{n+1}.$
...

Note that: with $a, b>0$,
Since
$ \frac{1}{a}+\frac{1}{b} = \frac{4}{a+b} \iff a=b.$
Thanks from quasi

Last edited by The Epsilon; January 16th, 2017 at 10:39 PM.
The Epsilon is offline  
Reply

  My Math Forum > College Math Forum > Real Analysis

Tags
show


« Taylor series | - »

Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Show that fahad nasir Trigonometry 2 October 2nd, 2016 08:51 AM
show it mared Geometry 0 May 29th, 2015 11:48 AM
show mared Geometry 7 May 16th, 2015 09:50 PM
want to show that show that two infinite summations R equal notnaeem Real Analysis 4 August 16th, 2010 12:32 PM
Show that fog is one-to-one 450081592 Calculus 2 January 25th, 2010 10:38 AM





Copyright © 2017 My Math Forum. All rights reserved.