My Math Forum Proof of divergence for the harmonic series

 Algebra Pre-Algebra and Basic Algebra Math Forum

April 18th, 2012, 11:05 AM   #11
Math Team

Joined: Dec 2006
From: Lexington, MA

Posts: 3,267
Thanks: 408

Re: Proof of divergence for the harmonic series

Hello, SlamDunk!

I'll give the proof that I was shown many years ago.

Quote:
 Prove that the Harmonic Series diverges.

$\text{W\!e have: }\:S \;=\;1\,+\,\left(\frac{1}{2}\right)\,+\,\left(\fra c{1}{3}\,+\,\frac{1}{4}\right)\,+\,\left(\frac{1}{ 5}\,+\,\frac{1}{6}\,+\,\frac{1}{7}\,+\,\frac{1}{8} \right)\,+\,\cdots$

$\text{Now we run a comparison . . .}$

[color=beige]. . [/color]$\begin{array}{cccccccccc}1 &&=&& 1 &&=&& 1 \\ \\ \\
\frac{1}{2} &&=&& \frac{1}{2} &&=&& \frac{1}{2} \\ \\ \\
\frac{1}{3}+\frac{1}{4} &&>&& \frac{1}{4}+\frac{1}{4} &&=&& \frac{1}{2} \\ \\ \\
\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8} &&>&& \frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8} &&=&& \frac{1}{2} \\ \\ \\
\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\,\cdots\,+\ frac{1}{16} &&>&& \frac{1}{16}+\frac{1}{16}+\,\cdots\,+\frac{1}{16} &&=&& \frac{1}{2} \\ \\ \\ \\ \\
\vdots && && \vdots && && \vdots \\ \\ \\
\end{array}$

$\text{The sum of the first column is the Harmonic Series}
\;\;\;\text{which is greater than the sum of the third column,}
\;\;\;\text{which is a divergent series: }\:1\,+\,\frac{1}{2}\,+\,\frac{1}{2}\,+\,\cdots$

$\text{Therefore, the Harmonic Series diverges.}$

April 18th, 2012, 11:21 AM   #12
Senior Member

Joined: Jul 2010
From: St. Augustine, FL., U.S.A.'s oldest city

Posts: 12,211
Thanks: 521

Math Focus: Calculus/ODEs
Re: Proof of divergence for the harmonic series

Quote:
Originally Posted by SlamDunk
Quote:
 Originally Posted by MarkFL If I were going to prove the original statement, I would write it in the form: $2+\frac{2}{n^2-1}>2$ $\frac{2}{n^2-1}>0$
How did you go from our original statement to this? I thought I understood it yesterday but I cannot seem to manipulate our given information into this.
Begin with:

$\frac{1}{n-1}+\frac{1}{n}+\frac{1}{n+1}>\frac{3}{n}$

Multiply through by n:

$\frac{n}{n-1}+1+\frac{n}{n+1}>3$

$\frac{n}{n-1}+\frac{n}{n+1}>2$

Combine on the left:

$\frac{n(n+1)+n(n-1)}{n^2-1}>2$

$\frac{2n^2}{n^2-1}>2$

$\frac{2n^2-2+2}{n^2-1}>2$

$\frac{2$$n^2-1$$+2}{n^2-1}>2$

$\frac{2$$n^2-1$$}{n^2-1}+\frac{2}{n^2-1}>2$

$2+\frac{2}{n^2-1}>2$

$\frac{2}{n^2-1}>0$

 Tags divergence, harmonic, proof, series

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post walter r Real Analysis 4 May 4th, 2013 07:34 PM aaron-math Calculus 5 December 17th, 2011 06:27 PM patient0 Real Analysis 5 December 11th, 2010 06:17 AM HammerTime Real Analysis 1 November 20th, 2008 04:18 AM brunojo Real Analysis 11 December 2nd, 2007 07:49 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top