If a series, such as
has a value , then the finite sums
that we can compute must eventually get close to .
Let us call the th partial sum. For the infinite series to converge to a value it is necessary that the sequence formed from the partial sums converges to some definite number, which is going to be the sum of the infinite series. Thus we see that convergence of a series is a special case of convergence of sequences. Therefore we start our discussion with sequences and return to series later. In fact, quite a bit later.
A sequence is a list of real numbers
given in a definite order
and indexed by natural numbers. Sometimes we may start
sequences with , sometimes with
, and sometimes with . The starting
point doesn't really matter. What matters is tha
is defined for all from this starting point
onwards. We will refer to the sequence as
when the starting point and
the choice of dummy variable are clear from context.
It is helpful to remember that is a dummy variable in
and could equally well have been some other
letter (with the same meaning) as in
Given a sequence of real numbers
we need to say what it means for to
get close to a number as
gets large. In other words, we need to define what it means for
the limit of the sequence to be as tends to
We imagine a conversation between a customer and a salesperson. The customer is thinking of buying a very expensive sequence which he is told converges to .
CUSTOMER: You've told me that as . So if I need my values of to be within a half of (i.e., ) what do I have to do?
SALESPERSON: Oh that's easy. Any value of will do that provided .
CUSTOMER: And what if I need ?
SALESPERSON: You get that provided is at least 102.
CUSTOMER: And ?
SALESPERSON: (Thinks for a few minutes) For all it will be true that .
CUSTOMER: I'm going to use this sequence a lot, and I have a lot of other numbers for which I need . I hope you don't mind, but I couldn't bring them all with me today to show you.
SALESPERSON: You'll need to tell me a little more. What sort of numbers are these ?
CUSTOMER: Oh sorry! Yes I should have said that these numbers are always positive real numbers. But they could be very small indeed.
SALESPERSON: (Looks at the box the sequence is in.)
In that case, I think this sequence is just what you need. Do you
see the guarantee that the manufacturers have written on it?
(They both read the small print on the box.
We want you to be 100% satisfied with your purchase of this
If you think there is a problem with your sequence, just send us a
small positive number to us and we will provide you with
a natural number and will guarantee to you that
holds for all .)
CUSTOMER: That's perfect. Just what I want.
The dialogue just given shows that there are a number of subtle issues
to do with convergence of sequences. First there is an
This must be a positive real number since otherwise
cannot possibly hold. Also will hold only if
and we are interested in sequences that converge to some number
but need not ever equal that number.
Given some such positive error term, the sequence must be very close to
, in particular having a distance from that is less than the allowed error.
But it's important to remember that it is not enough that our sequence just does this
once or twice; it must be within the error for all values from some point onwards.
holds for any
where is some natural number, the choice of which is allowed to depend
on . If we put this together we get the
official definition of convergence.
The sequence converges to as if: for all in the reals there is a natural number such that holds for all .
If we use the symbols
and to mean
implies, we can write this as:
the sequence converges to as if
If that seems quite complicated to you, you are right. It is quite complicated. But it really is the simplest possible definition (sigh). I've tried to explain where it comes from and why. There will be further motivation and reasons later. At this point, the best thing you can do is to learn this definition—either by recalling the dialogue, or by any other means. The symbolic form with , and is the most useful one to learn, for reasons that will become apparent soon.
Convergence to is a little easier than convergence to an arbitrary since if then can be written . So:
The sequence converges to zero as if
Sequences that converge to zero are called null sequences.
If we check through the examples to given earlier, we should be able to check that our new definition agrees with our original intuition. I won't do this here, but it could be done in lectures. Note that if the customer had been sold the sequence he would have been sold a dud! This is because, although the definition works for very small including , it doesn't work for . But then, the manufacturers wouldn't have been able to honour their guarantee, so our customer would at least have been able to get his money back if he had been following this course.