Drexel dragonThe Math ForumDonate to the Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

Sum of Harmonic Series

Date: 5/9/96 at 23:17:50
From: Paulette M. Tarantola
Subject: Sum of Harmonic Series

My question pertains to what the total sum in terms of variables is 
for the series:


I am having difficulty finding the sum and do not know were to start. 
I would appreciate any help.

Date: 11/13/96 at 08:04:38
From: Doctor Kuznicki
Subject: Re: Sum of Harmonic Series

Hi Paulette,

First simplify your series equation:

s = Sum 1/(n+1) 
[we know that this series diverges]

Now write out the series (as you did above) but include the n-1 part:

1/2 + 1/3 + 1/4 + ... + 1/n + 1/(n+1)

If you know the sum equation for one of these terms (1/n) then you can 
use this in your total sum:

 inf                                                   (n+1)
 Sum (1/n) = ln(n+1)  [since 1 + 1/2 + 1/3 +...+1/n > Integral 1/x dx]
 n=1                                                     1

Now use this in your equation:
[Hint: replace (1/2 + 1/3 + 1/4 + ... + 1/n) with (ln(n+1) - 1)]

 s = ln(n+1) + 1/(n+1) - 1

The easiest way to solving these problems is by finding ways to use 
already known divergent series sum equations.

One should also consider the other inequality:

1/2 + 1/3 + 1/4 + ... + 1/(n+1)  < Integral (1 to n+1) [1/x.dx] = 

So we have:

  ln(n+1) + 1/(n+1) -1 < 1/2 + 1/3 + .. + 1/(n+1) < ln(n+1)

-Doctor Kuznicki,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
Associated Topics:
High School Sequences, Series

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994-2013 The Math Forum