Convergence of Alternating Series

Date: 05/14/2000 at 20:55:45
From: Ki
Subject: Convergence of Alternating Series

Hi, 

I'm a little confused about convergence of alternating series. I've 
been told that the test for convergence of alternating series is:

     a_n > a_n+1   and   lim (n->inf) a_n = 0

But why not just check:

     lim (n->inf) a_n  = 0?

Date: 05/15/2000 at 03:47:57
From: Doctor Schwa
Subject: Re: Convergence of Alternating Series

Excellent question.

To think about it intuitively, if you just have the limit going to 
zero, the "wiggles" up and down in the sum (adding and subtracting) 
are getting smaller and smaller, but there's nothing to stop them from 
overall drifting upwards. The first condition ensures that you never 
get past any boundary once you set it.

More concretely, let's try to make a sequence that drifts up to 
infinity yet still has lim a_n = 0. We'll add BIG positive things that 
gradually get smaller, and SMALL negative things that also gradually 
get smaller. How about this:

     2/1 - 1/1 + 2/2 - 1/2 + 2/3 - 1/3 + 2/4 - 1/4 + ...

That is, twice the harmonic series, minus the harmonic series.

Note, by the way, that this condition is not an "if and only if" - 
that is, if it's true, the series converges, but it's possible for the 
series to converge if it's not true, too. You just need to use another 
test to do it.

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

Date: 05/15/2000 at 22:57:15
From: Ki
Subject: Re: Convergence of Alternating Series

Thank you very much for your reply. I've always thought that:

     lim (n->0)  a_n = 0

implied that at a certain term k, a_k > a_k+1. Are there any other 
tests of convergence specifically for alternating series?

Date: 05/16/2000 at 19:03:27
From: Doctor Schwa
Subject: Re: Convergence of Alternating Series

It implies that the sequence decreases most of the time, but it 
doesn't imply what you need for the alternating series test. For the 
alternating series test, you need that for some term n, from that term 
on it's ALWAYS decreasing. That is, for ALL k > n, a_k > a_k+1.

My last message gave one example of a sequence that has a limit of 0 
but is NEVER always decreasing, so the alternating series test 
wouldn't apply to its sum (and indeed it doesn't converge):

     2/1 - 1/1 + 2/2 - 1/2 + 2/3 - 1/3 + 2/4 - 1/4 + ...

Of course, they can converge, too:

     a_n = (-1)^n/n^2 (when n is not a perfect square), 
           (-1)^n/n   (if n is a perfect square)

I don't know of any other tests that work specifically for alternating 
series. The integral test, which is my favorite test in general, tends 
to be awkward with alternating series. Alternating series that aren't 
absolutely convergent, but do converge, are very tricky to work with!

If you want an example that shows just how tricky alternating series 
can be (or indeed any kind of series with a mixture of + and - signs), 
I'm happy to provide one or two of my favorites.

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

Date: 05/16/2000 at 22:05:18
From: Ki
Subject: Re: Convergence of Alternating Series

Thanks again for the reply. Yes, I'd be delighted if you could send me 
some.

Date: 05/17/2000 at 12:42:13
From: Doctor Schwa
Subject: Re: Convergence of Alternating Series

Here's my all-time favorite. You can think of variations on this one 
on your own.

Here's a simple series:

     1 - 1/2 - 1/4 - 1/8 - 1/16 - 1/32 ... = 0

and similarly

     0 +  1  - 1/2 - 1/4 - 1/8  - 1/16 ... = 0

and

     0 +  0 +   1  - 1/2 - 1/4  - 1/8  ... = 0

and

     0 +  0 +  0 +   1   - 1/2  - 1/4  ... = 0

and

     0 +  0 +  0 +   0  +   1   - 1/2  ... = 0

and

     0 +  0 +  0 +   0  +   0   +  1   ... = 0

and so on.

So, clearly, the sum of the whole thing is zero, right? Infinitely 
many zeros? Well, let's try adding vertically first.

     1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 ... = 2

So zero equals two?

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