Earlier in Definition 4.4.1 on Page 122 the notion of limit of a sequence was discussed. There is a very closely related concept called an infinite series which is dealt with in this section.

whenever the limit exists and is finite. In this case the series is said to converge. If it does not converge, it is said to diverge. The sequence
From this definition, it should be clear that infinite sums do not always make sense. Sometimes they do and sometimes they don’t, depending on the behavior of the partial sums. As an example, consider ∑ _{k=1}^{∞}
Example 5.1.2 Find the infinite sum, ∑ _{n=1}^{∞}
Note

Proposition 5.1.3 Let a_{k} ≥ 0. Then

When the sequence is not bounded above, ∑ _{k=m}^{∞}a_{k} diverges.
Proof: It follows

If it is bounded above, then by the form of completeness found in Theorem 4.9.6 on Page 161 it follows the sequence of partial sums converges to sup
In the case where a_{k} ≥ 0, the above proposition shows there are only two alternatives available. Either the sequence of partial sums is bounded above or it is not bounded above. In the first case convergence occurs and in the second case, the infinite series diverges. For this reason, people will sometimes write ∑ _{k=m}^{∞}a_{k} < ∞ to denote the case where convergence occurs and ∑ _{k=m}^{∞}a_{k} = ∞ for the case where divergence occurs. Be very careful you never think this way in the case where it is not true that all a_{k} ≥ 0. For example, the partial sums of ∑ _{k=1}^{∞}
One of the most important examples of a convergent series is the geometric series. This series is ∑ _{n=0}^{∞}r^{n}. The study of this series depends on simple high school algebra and Theorem 4.4.9 on Page 133. Let S_{n} ≡∑ _{k=0}^{n}r^{k}. Then

Therefore, subtracting the second equation from the first yields

and so a formula for S_{n} is available. In fact, if r≠1,

By Theorem 4.4.9, lim_{n→∞}S_{n} =
Theorem 5.1.4 The geometric series, ∑ _{n=0}^{∞}r^{n} converges and equals
If the series do converge, the following holds.
Theorem 5.1.5 If ∑ _{k=m}^{∞}a_{k} and ∑ _{k=m}^{∞}b_{k} both converge and x,y are numbers, then
 (5.1) 
 (5.2) 
 (5.3) 
where in the last inequality, the last sum equals +∞ if the partial sums are not bounded above.
Proof: The above theorem is really only a restatement of Theorem 4.4.6 on Page 125 and the above definitions of infinite series. Thus

To establish 5.2, use Theorem 4.4.6 on Page 125 to write

and so

Recall that if lim_{n→∞}A_{n} = A, then lim_{n→∞}
Example 5.1.6 Find ∑ _{n=0}^{∞}
From the above theorem and Theorem 5.1.4,
The following criterion is useful in checking convergence. All it is saying is that the series converges if and only if the sequence of partial sums is Cauchy. This is what the given criterion says. However, this is not new information.
 (5.4) 
Proof: Suppose first that the series converges. Then
 (5.5) 
Next suppose 5.4 holds. Then from 5.5 it follows upon letting p be replaced with p + 1 that