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### 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
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/
```
Associated Topics:
High School Sequences, Series

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