How do we ``add up" a countable infinity of numbers? What meaning can we give to infinite sums such as and ? To answer these questions we first need to look at the concept of a sequence.
A sequence is a function whose domain is the set of positive integers. The functional values, the , are called the terms of the sequence. (For further details, see the text.) We say (informally) that has a limit L if, as n gets larger and larger, all the terms of the sequence are very close to L. A sequence that has a limit is said to converge. A sequence that does not have a limit is said to diverge.
If L is the limit of , then what do you think is the limit of the sequence ? If we are given and we suspect that L is its limit, we can use to test if L is indeed the limit. Think about how this process works; consult page 492 of the text.
An important sequence theorem (one that we will use many times) says that a bounded monotone sequence must converge. Much of the time we will use this theorem in the context of an increasing sequence ( for all k) that is bounded above (there exists a number B such that for all k). In this context the theorem says that an increasing sequence that is bounded above must converge.
Now we return to the question of what meaning can be given to the infinite series .
The procedure involves defining to be and looking at , the sequence of partial sums. If, as n goes to infinity, this sequence has a limit S, we define to be S and say that the infinite series converges to S. In other words, we use finite sums and a limit process to give meaning to the sum of a countable infinity of numbers.
The good news is that this process seems fairly easy for one who understands sequences - just find as a function of n and then find . The bad news is that most of the time this process breaks down because it is impossible to find a formula for as a function n. Fortunately we have available to us a variety of tests that can be used to tell if an infinite series converges (even though they don't tell us the value to which it converges, when it does converge). If a series converges, then - for an appropriate value of n - we use the number as an approximation to the actual sum S.