If a series, such as
has a value , then the finite sums
that we can compute must eventually get close to .
Let us call
it is necessary that
the sequence (
Definition.
A sequence is a list of real numbers
_{1,}_{2,}_{3,
}
given in a definite order
and indexed by natural numbers. Sometimes we may start
sequences with _{0}, sometimes with
_{1}, and sometimes with _{27}. The starting
point doesn't really matter. What matters is tha _{
}
is defined for all the sequence (_{
)
=1,2,
}
or just the sequence (_{
)
}
when the starting point and
the choice of dummy variable the sequence (_{
)
}
and could equally well have been some other
letter (with the same meaning) as in
the sequence (_{
)
}
.
Given a sequence of real numbers (_{
)
}
we need to say what it means for _{
} to get close to a number
. In other words, we need to define what it means for
the limit of the sequence (_{
)
} to be
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
SALESPERSON: Oh that's easy. Any value of _{
} will do that
provided
CUSTOMER: And what if I need
SALESPERSON: You get that provided
CUSTOMER: And
SALESPERSON: (Thinks for a few minutes)
For all
CUSTOMER: I'm going to use this sequence a lot, and I have a lot of other
numbers
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
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.
It says: We want you to be 100% satisfied with your purchase of this
sequence (_{
)
}.
If you think there is a problem with your sequence, just send us a
small positive number
)
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 error term
where
Definition.
If we use the symbols
for all
,
there exists
,
and implies
, we can write this as:
the sequence (_{
)
} converges to
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
Convergence to 0 is a little easier than convergence
to an arbitrary
Definition.
The sequence (_{
)
} converges to zero as
Sequences that converge to zero are called null sequences.
If we check through the examples (_{
)
} to (