This web page starts a new section of the course: infinite series. An infinite series is an expression such as and we are going to learn how to understand and work with such series rigorously.
This page starts with the definition of convergence for infinite series and three basic but very important results about them. Examples of convergent series will appear in a later page.
An infinite series is an expression of the form
is a sequence. We shall say this series converges to
if the sequence formed from
converges to , i.e., as . The term is called the th partial sum of . We say the series converges if it converges to some .
The series converges to . To see this, write and observe that . Thus as by the continuity of and the standard null sequence .
The series does not converge. To see this, write and note that is if is odd and if is even. The sequence does not converge so neither does the series .
We shall prove three very important but straightfoward results about
series to get our theory off the ground. The first of these states that,
as with sequences, the convergence of
on the behaviour of as
goes to infinity, and
not on the first few terms—where
few means any finite number
of terms, such as 1000000000. In other words, to show that
converges we can ignore
any finite number of terms
and just need to prove converges.
Proposition on eventual convergence, or ignoring finitely many terms.
Suppose and the series converges. Then the series also converges.
Let , and let . Then for , where is the partial sum . But converges, so for some . So by the convergence of the constant sequence and the continuity of , and hence converges.
This result is similar to the very easy one for sequences that says: if the subsequence consisting of all terms in after the th converges then the whole sequence converges. The proposition on eventual convergence is rather useful, however, as it shows that when investigating the convergence of a series, we can always ignore the (perhaps erratic) behaviour of finitely many terms at the beginning.
The next result is commonly used in its contrapositive form to show that a series does not converge. It cannot be used to show that a series does converge.
Null sequence test.
Suppose the series converges. Then as .
Let be the th partial sum, and suppose as . Then as , by continuity. Therefore is a null sequence and hence so is .
The series does not converge.
The sequence does not converge, by a previous result in the course, so in particular does not converge to . Hence by the null sequence test does not converge.
Our final result here is an application of the monotone convergence theorem to series, and will be used implicitly or explicitly a huge amount in the section on series.
Suppose the series consists of nonnegative terms only. Then the sequence of partial sums, , is monotonic nondecreasing and hence the series converges if and only if the sequence of partial sums is bounded.
By the proposition on eventual convergence, the monotonicity theorem also applies to any series with only finitely many negative terms: such a series is convergent if and only if the sequence of partial sums is bounded.