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:

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

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:

    inf
s = Sum 1/(n+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] = 
ln(n+1)

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/