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Series Types

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

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