Convergence of an Alternating Series

Date: 02/23/98 at 21:57:19
From: Harris Welt
Subject: convergence of an alternating series

Dear Dr. Math,

I'm trying to figure out whether or not this series is convergent.

Computing the sum from n = 1 to infinity, (-1)^(n+1)*(ln (n)/n), is 
the series convergent if the limit of (ln (n)/n) is convergent?  
If so, how can this limit be found?  If not, why not?  Also, if you 
get a chance, explain why this works.  I've been having a lot of 
trouble understanding my math teacher in this area.

Thanks,
Harris

Date: 02/24/98 at 09:34:25
From: Doctor Anthony
Subject: Re: convergence of an alternating series

For an alternating series, you will ALWAYS have convergence if the 
limit of u(r) as r -> infinity is zero.  So this series converges if:

     Lt        ln(n)/n = 0 
    n->infin.

    Using l'Hopital's rule as n->infinity

       ln(n)      1/n
      ------  =  ------  = 1/n   -> 0  as n -> infinity. 
         n         1

You can see why the alternating series converges if you plot S1, S2, 
S3, etc on a number line. S1 is positive and goes to the right. S2 is 
the result of subtracting a term from S1 so brings you to the left a 
bit, S3 is the result of adding a term to S2 so takes you to the right 
again, but not as far as S1.  The points will jump to and fro about 
the final position, S, which is less than S1 but greater than S2.


                                 |
  0-------------S2------S4---S6--|--S7---S5----S3---------S1
                                 |
                                 S


-Doctor Anthony,  The Math Forum
 http://mathforum.org/dr.math/