Series Types

Date: 05/11/97 at 00:36:44
From: sam
Subject: Convergent, oscillating and divergent series and how to 
obtain the first four Maclaurin series expansions for the series y=e^x

What are the definitions of convergent, divergent, and oscillating 
series? I don't have much of an idea as to what they are. Can you 
please help?

Sam

Date: 05/11/97 at 11:57:23
From: Doctor Anthony
Subject: Re: Convergent, oscillating and divergent series and how to 
obtain the first four Maclaurin series expansions for the series y=e^x

Dear Sam, 

A convergent series is one which tends to a finite value as you 
increase the number of terms to infinity.

An example of a convergent series is: 1 + 1/2 + 1/4 + 1/8 + 1/16 + ...

This is a geometric series with common ratio = 1/2, and the sum to 
infinity is 2. So the sum of an infinite number of terms of this 
series will not exceed 2.

A divergent series will become infinite if you take enough terms.  
A good example is: 1 + 1/2 + 1/3 + 1/4 + 1/5 +.... to infinity.  
If you take a sufficient number of terms, the sum will increase 
without limit.

An oscillating series will produce two different results depending on 
whether you take an even or odd number of terms.

Example 1 - 1 + 1 - 1 + 1 - 1 + .......

For instance, if we add the first four terms together, we see that the 
sum is 0.  But if we add the first three terms together, the sum is 1.  

Incidentally, the e^x series is convergent for all finite vales of x.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/