文档介绍：Section
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Infinite Series:
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In this section we continue our discussion of the convergence properties of infinite series.
Now that we have two classes of series, namely, the geometric series and the p-series, for
which classification as either convergent or divergent is relatively easy, it is reasonable to
develop tests for convergence based paring a given series with a series of known
behavior. We will see this idea first in parison test, which we will later generalize
with the parison test.
∞∞
To begin, suppose an is a convergent series with an ≥ 0 for all n and bn
Pn=1 Pn=1
is a series with 0 ≤ bn ≤ an for all n. Let
∞
L = a .
X n
n=1
∞∞
If sn is the nth partial sum of an and tn is the nth partial sum of bn, then
Pn=1 Pn=1
tn ≤ sn ()
for n = 1, 2, 3, . . .. Since an ≥ 0 for all n, the sequence {sn} is increasing; hence
sn ≤ L ()
for all n. Since bn ≥ 0, n = 1, 2, 3, . . ., {tn} is also an increasing sequence which, by
∞
() and (), is bounded above by L. Hence lim tn exists, showing that bn
n→∞ Pn=1
converges. Moreover, from (),
∞∞
b ≤ a . ()
X n X n
n=1 n=1
∞∞
Now suppose that an is a divergent series with an ≥ 0 for all n and bn is
Pn=1 ∞ Pn=1
a series with an ≤ bn for all n. If sn is the nth part