Defining a SequenceDate: 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 n->infinity 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 n->infinity 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: n S(n) = SUM x(i) i=1 limit S(n) = S n->infinity 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 http://mathforum.org/dr.math/ |
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