What is
You may have encountered sequences long ago in earlier courses and
they seemed very difficult. You know much more mathematics now, so
they will probably seem easier. On the other hand,
we're going to go very quickly.
We will completely skip several topics from Chapter 11.
I will try to make what we skip clear. Note that the homework
has been modified to reflect the omitted topics.

A sequence is an ordered list of numbers. These numbers may be real,
complex, etc., etc., but in this book we will focus entirely on
sequences of real numbers. For example,
Since the sequence is ordered, we can view it as a function
with domain the natural numbers
.
Definition 6.1.1 (Sequence)
A
sequence is a function
that takes a natural number
to
. The number
is the
th term.
For example,
which we write as
.
Here's another example:
Example 6.1.2
The Fibonacci sequence
is defined recursively as follows:
for $n &ge#geq;3$
Let's return to the sequence
.
We write
, since the terms get arbitrarily
small.
Definition 6.1.3 (Limit of sequence)
If
is a sequence then that
sequence converges to
,
written
, if
gets arbitrarily close to
as
get sufficiently large.
SECRET RIGOROUS DEFINITION: For every
there
exists
such that for
we have
.
This is exactly like what we did in the previous course when we considered
limits of functions. If is a function, the meaning
of
is essentially the same. In fact, we have
the following fact.
As a corollary, note that this implies that all the facts about limits
that you know from functions also apply to sequences!
Example 6.1.6
The converse of Proposition
6.1.4 is false
in general,
i.e., knowing the limit of the sequence converges doesn't imply
that the limit of the function converges.
We have
, but
diverges.
The converse is OK if the limit involving the function converges.
Example 6.1.7
Compute
.
Answer:
.
William Stein
20060315