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 £© Section
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 £© Infinite Series Revisited
Recall from Section that for a given sequence {an}, the sequence {sn} with nth term
sn = a1 + a2 + a3 + · · · + an ()
is called an infinite series. An individual term sn is called a partial sum and we say the
series is convergent, or has a sum, if lim sn exists. If the series is not convergent, we say
n→∞
it is divergent. We write
∞
lim sn = a1 + a2 + a3 + · · · + an + · · · = an. ()
n→∞ X
n=1
n
Example In Section we saw that if an = r , n = 0, 1, 2, . . ., then the associated
infinite series, called a geometric series, is convergent if and only if −1 < r < 1, in which
case ∞
1
rn = . ()
X 1 − r
n=0
For example,
∞
1 1 3
= = .
X 3n 1 − 1 2
n=0 3
Geometric prise one of the few classes of series for which we can evaluate sums
exactly. For most series we can only approximate the sum puting the partial sums
sn for sufficiently large values of n. However, before this procedure es meaningful, we
must first know that the series converges. Hence, in this section, as well as in Sections ,
, and , one of our primary goals will be the development of methods for determining
whether a given series converges or diverges.
We begin by considering several basic properties of infinite series. First, suppose we
∞∞
know that both an and bn are convergent series with
Pn=1 Pn=1
∞
a = L
X n
n=1
and ∞
b = M.
X n
n=1
1
2 Infinite Series Revisited Section
∞∞
If sn is the nth partial sum of an, tn is the nth partial sum of bn, and un is
∞ Pn=1 Pn=1
the nth partial sum of (an + bn), then un = sn + tn. Thus
Pn=1
lim un = lim (sn + tn) = lim sn + lim tn = L + M. ()
n→∞ n→∞ n→∞ n→∞
That is,
∞∞∞
(a + b ) = a + b . ()
X n n X n X n
n=1 n=1 n=1
∞∞∞
Proposition If an and bn both converge, then (an + bn) converges
Pn=1 Pn=1 Pn=1
and ∞∞∞