Math 152 – Spring 2016
Section 10.2
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Section 10.2 – Series
Definition. If we add the terms of an infinite sequence {an }∞
n=1 , then we get an
infinite series
a1 + a2 + a3 + · · · + an + · · · .
We use the notation
∞
X
an
or
X
an
n=1
Question. How do we add infinitely many things? Consider the following two series:
1.
∞
X
n
We can define partial sums – sums of the first i terms denoted si :
n=1
s1 = 1
s2 = 1 + 2 = 3
s3 = 1 + 2 + 3 = 6
s4 = 1 + 2 + 3 + 4 = 10
..
.
sn = 1 + 2 + · · · + n =
n(n + 1)
2
..
.
This series clearly has an infinite sum – as we add more and more terms the sum
approaches infinity. We say the series is divergent.
2.
∞
X
1
.
2n
n=1
The partial sums si are
1
2
1
3
1
s2 = + 2 =
2 2
4
1
1
1
7
s3 = + 2 + 3 =
2 2
2
8
1
1
1
1
15
s4 = + 2 + 3 + 4 =
2 2
2
2
16
..
.
1
1
1
1
sn = + 2 + · · · + n = 1 − n
2 2
2
2
..
.
s1 =
As n → ∞ the partial sums approach 1. Therefore,
∞
X
1
= 1.
2n
n=1
Math 152 – Spring 2016
Definition. Given a series
Section 10.2
∞
X
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ai = a1 + a2 + a3 + · · · , let sn denote its nth partial
i=1
sum:
sn = a1 + a2 + · · · an =
n
X
ai
i=1
If the sequence {sn } is convergent and lim sn = s exists as a real number, then the
series
∞
X
n→∞
ai is called convergent and we write
i=1
a1 + a2 + a3 + · · · = s
or
∞
X
an = s
n=1
The number s is called the sum of the series. If the sequence {sn } does not converge,
then the series is called divergent.
Note. In the previous definition, it is easy to confuse what is a series and what is a
sequence.
• Series – the infinite sum
∞
X
an
n=1
• Sequence – the partial sums {sn } form a sequence where each sn =
n
X
ai .
i=1
The definition tells us that (provided the limit exists) the infinite series equals the limit
of the partial sums:
∞
n
X
X
ai = lim
ai = lim sn
i=1
n→∞
i=1
n→∞
Example 1. Determine if the following sequences converge or diverge. If they converge,
find the sum.
(a)
∞
X
n
n=1
(b)
∞
X
1
2n
n=1
Math 152 – Spring 2016
Section 10.2
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Geometric Series. A geometric series is a series of the form
a + ar + ar2 + ar3 + · · · + arn + · · · =
∞
X
arn−1
r 6= 0
n=1
where a and r 6= 0 are constants.
Question. For what values of r does a geometric series converge?
Theorem. The geometric series
∞
X
arn−1 = a + ar + ar2 + ar3 + · · · + arn + · · ·
n=1
is convergent if |r| < 1 and its sum is
∞
X
n=1
arn−1 =
a
1−r
for
|r| < 1.
If |r| ≥ 1, the geometric series is divergent.
Example 2. Determine if the series is convergent or divergent. If it is convergent, find
its sum.
(a)
∞
X
n=1
2−3n 5n+2
Math 152 – Spring 2016
(b)
∞
X
Section 10.2
sinn x, where −π/2 < x < π/2
n=1
(c)
∞
X
1
n(n
+ 1)
n=1
(d) 2 −
7
3
+
49
18
−
343
108
+ ···
Example 3. Write the number 2.15 as a fraction.
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Math 152 – Spring 2016
Section 10.2
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Harmonic Series. The harmonic series is the series
∞
X
1
1 1 1
1
= 1 + + + + ··· + + ···
n
2
3
4
n
n=1
Show that the harmonic series is divergent.
n
Terms: an
Partial Sums: sn
1
a1 = 1
s1 = 1
2
a2 =
1
2
s2 = 1.5
3
a3 =
1
3
s3 = 1.83
4
a4 =
1
4
s4 = 2.08
5
a5 =
1
5
s5 = 2.28
6
a6 =
1
6
s6 = 2.45
7
a7 =
1
7
s7 = 2.59
8
a8 =
1
8
s8 = 2.72
9
a9 =
1
9
s9 = 2.83
1
10
s10 = 2.93
10
a10 =
Theorem. If the series
∞
X
n=1
Sketch of proof:
an is convergent, then lim an = 0.
n→∞
Math 152 – Spring 2016
Section 10.2
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Test for Divergence. If lim an does not exist or lim an 6= 0, then the series
n→∞
n→∞
∞
X
an
n=1
is divergent.
Note 1: For
∞
X
an we have the following sequences and series:
n=1
• Series – the infinite sum
∞
X
an
n=1
• Sequence – the partial sums {sn } form a sequence where each sn =
n
X
ai .
i=1
• Sequence – the terms of the sum form a sequence {an } (this is the sequence
{a1 , a2 , a3 , . . .} used in the Test for Divergence and the terms are NOT being
added.)
Note 2: If lim an = 0, then the series might be convergent or it might be divergent
n→∞
– the test for divergence DOES NOT tell us ANYTHING in this case. Recall the har∞
X
1
1
= 0, but
is divergent.
monic series: lim
n→∞ n
n
n=1
Example 4. Determine if the following series are convergent or divergent. If a series
is convergent, find its sum.
(a)
∞
X
n=1
(b)
√
3n
n2 + 3
∞
X
n
ln n
n=1
Math 152 – Spring 2016
Theorem. If
∞
X
Section 10.2
an and
n=1
∞
X
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bn are convergent series, then the following series are
n=1
convergent and equal to what is listed below:
•
∞
X
can = c
n=1
•
∞
X
∞
X
(an + bn ) =
n=1
•
∞
X
an
n=1
∞
X
an +
n=1
(an − bn ) =
n=1
∞
X
n=1
∞
X
bn
n=1
an −
∞
X
bn
n=1
Example 5. Determine if the following series are convergent or divergent. If the series
is convergent, find its sum.
∞ X
1
2−3n 5n+2 − n
(a)
2
i=1
(b)
∞
X
3
n(n
+ 1)
n=2
Math 152 – Spring 2016
(c)
∞
X
ln n − ln(n2 − 3)
n=1
(d)
∞
X
e−n − e−n+1
n=1
n
∞
X
π
1
7
− n+1
(e)
32n
2
n=1
Section 10.2
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