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Defining a Sequence

Date: 11/04/98 at 00:35:54
From: Brian Bell
Subject: Sequence, series

Can you give me a good definition of sequence, series, convergence, 
and divergence, and explain how they all correlate to one another?

Thanks, Brian

Date: 11/04/98 at 12:36:47
From: Doctor Rob
Subject: Re: Sequence, series

A sequence is a function whose domain is the natural numbers, x:N -> C. 
(The range can be any set. We will restrict our attention to the 
complex numbers C.) Each image x(n) is called a term of the sequence.

A series S is the formal sum of all the terms of a sequence:

   S = x(1) + x(2) + x(3) + x(4) + ... + x(n) + ...

A sequence x converges to a limit L if, for any epsilon > 0, there 
exists a natural number N such that n > N implies that the distance 
from x(n) to L is less than epsilon. (In other words, all the terms of 
the sequence from point N on are close to L. You tell me how close you 
want to get and stay [within epsilon], and I will tell you how far 
down the sequence you have to go to make that happen [beyond the Nth 
term].) This is written:

      limit   x(n) = L

If a sequence x does not converge to any limit L, it is said to 
diverge. This is written as:

      limit   x(n) does not exist

A series is said to converge to a limit S if the sequence of partial 
sums of the terms converges to S.  A partial sum is:

   S(n) = SUM x(i)

      limit   S(n) = S

If a series does not converge to any limit S, it is said to diverge.

Example: One sequence x:N -> C is defined by x(n) = 1/n for all n in N.

Then x converges to the limit 0 because, if you give me epsilon > 0, 
I can pick N > 1/epsilon, and then if n > N > 1/epsilon > 0, then 
1/n < epsilon, so the distance between 1/n and 0 is less than epsilon.

The corresponding series is:

   S = x(1) + x(2) + x(3) + x(5) + ... + x(n) + ...
     = 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... + 1/n + ...

The sequence of partial sums is:

   1, 1+1/2, 1+1/2+1/3, 1+1/2+1/3+1/4, 1+1/2+1/3+1/4+1/5, ...  or
   1, 3/2, 11/6, 25/12, 137/60, 49/20, ...  or
   1, 1.500000, 1.833333, 2.083333, 2.283333, 2.450000, ...

This sequence of partial sums does not have a limit, because they get
arbitrarily large.  (This may not be obvious, but it is not too hard 
to prove.)  Therefore the series above diverges.

I hope this is what you had in mind.

- Doctor Rob, The Math Forum
Associated Topics:
High School Sequences, Series

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