Calculus II
Stewart
Dr. Berg
Spring 2010
Page 1
12.2
12.2
Series
If we try to add the elements of an infinite sequence
{
a
k
}
we get the
infinite
series
a
k
k
=
1
∞
∑
=
a
k
∑
=
a
1
+
a
2
+…+
a
k
+…
. Can we actually add infinitely many
numbers? In Zeno’s most famous paradox, Achilles can never catch the tortoise, if the
tortoise is given a head start, because the Greeks assumed that such a sum must be
infinite. The argument is that, when Achilles reaches the place where the tortoise started,
the tortoise has moved ahead to a new location. When Achilles reaches that location, the
tortoise has moved ahead again, and so on, ad infinitum.
Definition
a
k
k
=
1
∞
∑
≡
lim
n
→∞
a
k
k
=
1
n
∑
=
lim
n
→∞
S
n
=
S
where
S
n
=
a
k
k
=
1
n
∑
is the
n
th partial sum
of the
series. If the limit exists, we say the series
converges
. Otherwise, it
diverges
.
S
is called
the
sum
of the series.
Note:
This is actually the limit of a sequence
of partial sums
S
n
{
}
. It is also common to
begin the sequence at
k
=
0
, or even a negative number.
Examples A
1
k
(
k
+
1)
k
=
1
∞
∑
=
1
k
−
1
k
+
1
k
=
1
∞
∑
is convergent since
S
n
=
1
1
−
1
2
+
1
2
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 Spring '10
 ZHENG
 Calculus, Dr. Berg, Calculus IIStewart

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