
NOTE 2
The converse of Theorem 6 is not true in general. If
, we can
not conclude that
is convergent.
Observe that for the harmonic series
we have
as
, but we showed in Example 8 that
is divergent.
Test for Divergence
If
does not exist or if
, then the
series
is divergent.
The Test for Divergence follows from Theorem 6 because, if the series is not divergent,
then it is convergent, and so
.
Show that the series
diverges.
SOLUTION
So the series diverges by the Test for Divergence.
NOTE 3
If we find that
, we know that
is divergent. If we find that
, we know
nothing
about the convergence or divergence of
. Remember
the warning in Note 2: If
, the series
might converge or it might diverge.
Theorem
If
and
are convergent series, then so are the series
(where
is a constant),
, and
, and
(i)
(ii)
(iii)
These properties of convergent series follow from the corresponding Limit Laws for
Sequences in Section 11.1. For instance, here is how part (ii) of Theorem 8 is proved:
Let
s
n
lim
n
l
s
n
s
n
1
l
n
l
lim
n
l
s
n
1
s
lim
n
l
a
n
lim
n
l
s
n
s
n
1
lim
n
l
s
n
lim
n
l
s
n
1
s
s
0
a
n
s
n
a
n
a
n
s
n
s
a
n
lim
n
l
a
n
0
a
n
1
n
a
n
1
n
l
0
n
l
1
n
7
lim
n
l
a
n
lim
n
l
a
n
0
n
1
a
n
lim
n
l
a
n
0
n
1
n
2
5
n
2
4
lim
n
l
a
n
lim
n
l
n
2
5
n
2
4
lim
n
l
1
5
4
n
2
1
5
0
lim
n
l
a
n
0
a
n
lim
n
l
a
n
0
a
n
lim
n
l
a
n
0
a
n
8
a
n
b
n
ca
n
c
a
n
b
n
a
n
b
n
n
1
ca
n
c
n
1
a
n
n
1
a
n
b
n
n
1
a
n
n
1
b
n
n
1
a
n
b
n
n
1
a
n
n
1
b
n
EXAMPLE 9
t
n
1
b
n
t
n
n
i
1
b
i
s
n
1
a
n
s
n
n
i
1
a
i
a
n
a
n
s
n
s
n
1
s
n
a
1
a
2
a
n
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CHAPTER 11
The
n
th partial sum for the series
is
and, using Equation 4.2.10, we have
Therefore
is convergent and its sum is
Find the sum of the series
.
SOLUTION
The series
is a geometric series with
and
, so
In Example 7 we found that
So, by Theorem 8, the given series is convergent and
NOTE 4
A finite number of terms doesn’t affect the convergence or divergence of a
series. For instance, suppose that we were able to show that the series
is convergent. Since
it follows that the entire series
is convergent. Similarly, if it is known that
the series
converges, then the full series
is also convergent.
u
n
n
i
1
a
i
b
i
lim
n
l
u
n
lim
n
l
n
i
1
a
i
b
i
lim
n
l
n
i
1
a
i
n
i
1
b
i
lim
n
l
n
i
1
a
i
lim
n
l
n
i
1
b
i
lim
n
l
s
n
lim
n
l
t
n
s
t
a
n
b
n
n
1
a
n
b
n
s
t
n
1
a
n
n
1
b
n
a
n
b
n
n
1
3
n n
1
1
2
n
1 2
n
a
1
2
r
1
2
n
1
1
2
n
1
2
1
1
2
1
n
1
1
n n
1
1
n
1
3
n n
1
1
2
n
3
n
1
1
n n
1
n
1
1
2
n
3
1
1
4
n
4
n
n
3
1
n
1
n
n
3
1
1
2
2
9
3
28
n
4
n
n
3
1
n
1
n
n
3
1
n
N
1
a
n
n
1
a
n
N
n
1
a
n
n
N
1
a
n
EXAMPLE 10
SERIES
735
1.
(a)
What is the difference between a sequence and a series?
(b) What is a convergent series? What is a divergent series?
2.
Explain what it means to say that
.
3–4
Calculate the sum of the series
whose partial sums
are given.
3.
4.
5–8
Calculate the first eight terms of the sequence of partial
sums correct to four decimal places. Does it appear that the series
is convergent or divergent?