Until the final section of this chapter, we will restrict our attention to series with an≥ 0. Thus the partial sums are increasing:
If the series
an is to converge, there must be some
B such that
sn≤B for all
n, or the
sn will become arbitrarily large. Such a
B is called
an upper bound. The value to which the series converges is the least of all possible
upper bounds. It turns out that whenever the sequence
{sn} of partial sums has an
upper bound, there exists a least upper bound, to which the series converges. This fact
enables us to prove the comparison test, stated below.
For two series
an,
bn, with
an, bn≥ 0 for all
n, suppose there exists a number
C > 0 such that
To prove this statement, it suffices to show that the number
Cbn is
a bound for the partial sums
a1 + a2 + ... + an. Then the least upper bound of these
partial sums must exist and is clearly less than or equal to
Cbn.
Thus we need only note that
A similar test enables us to show that certain series diverge. If
an
and
bn are again two series with
an, bn≥ 0 for all
n,
suppose that there exists
C≥ 0 such that
an≥Cbn for all
n and that
bn diverges. Then
an also diverges. The
proof of this fact is similar to the previous proof--since the partial sums of the
bn
become arbitrarily large and
an≤C
