section 8.2: series practice hw from stewart textbook (not to hand in) p. 575 # 9-15 odd, 19, 21,...
TRANSCRIPT
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Section 8.2: Series
Practice HW from Stewart Textbook
(not to hand in)
p. 575 # 9-15 odd, 19, 21, 23, 25, 31, 33
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Infinite Series
Given an infinite sequence , then
is called an infinite series.
}{ na
1321
nnn aaaaa
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Note
is the infinite sequence
is an infinite series.
1n
nan ,
5
4 ,
4
3 ,
3
2 ,
2
1
5
4
4
3
3
2
2
1
111 nnn n
na
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Consider
Note that is a sequence of
numbers called a sequence of partial sums.
nn aaaaS
aaaS
aaS
aS
321
3213
212
11
nSSS ,,, 21
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Definition
For an infinite series , the partial sum is
given by
If the sequence of partial sums converges to S,
the series converges.
na thn
nn aaaaS 321
1nna
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The limit S is the sum of the series
If the sequence diverges, then the series
diverges.
121
)(limlim
nnn
nn
naaaaSS
}{ nS
1nna
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Example 1: Find the first five sequence of partial
sum terms of the series
Find a formula that describes the sequence of
partial sums and determine whether the
sequence converges or diverges.
Solution:
1n
n
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Example 2: Find the first five sequence of partial
sum terms of the series
Find a formula that describes the sequence of
partial sums and determine whether the
sequence converges or diverges.
Solution:
1 16
1
8
1
4
1
2
1
2
1
nn
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Geometric Series
A geometric series is given by
with ratio r.
1
21
n
nn arararaar
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Notes
1. The geometric series converges if and only if
. When , the sum of the series (the value the series converges to) is
If , then the geometric series diverges.
1|| r 1|| r
1
1
1n
n
r
aar
1|| r
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2. The value a is the first term of the series.
3.The ratio r is the factor you multiply the previous term by to get the next one. That is,
term)1(
termor term)1( term)(
th
ththth
n
nrnnr
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Example 3: Determine whether the series
is convergent or divergent. If convergent,
find its sum.
Solution:
1
1
3
22
n
n
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Example 4: Determine whether the series
is convergent or divergent. If
convergent, find its sum.
Solution:
9
1
3
113
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Example 5: Determine whether the series
is convergent or divergent. If convergent,
find its sum.
Solution:
1
15 2n
nn
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Properties of Series (p. ??)
111
)(n
nn
nn
nn baba
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Example 6: Determine whether the series
is convergent or divergent.
If convergent, find its sum.
Solution:
1
11 3
2
2
1
nnn
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Applications of Geometric Series
Example 7: Express as a
ratio of integers.
Solution: (In typewritten notes)
73737373.037.0
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Tests For Non-Geometric Series
Most series are not geometric – that is, there is
not a ratio r that you multiply each term to get to
the next term. We will be looking at other ways to
determine the convergence and divergence of
series in upcoming sections.
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Some other ways to test series
1. Divergence Test: If the sequence does not converge to 0, then the series diverges.
Note: This is only a test for divergence – if the sequence converges to 0 does not necessarily mean the series converges.
}{ na
na
na}{ na
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2.Examine the partial sums to determine convergence or divergence (Examples 1 and 2
of this section).
3.Techniques discussed in upcoming sections.
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Example 8: Demonstrate why the series
is not geometric. Then
analyze whether the series is convergent or
divergent.
Solution:
11
4
9
3
7
2
5
1
321n n
n
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Example 9: Analyze whether the series
is convergent or divergent.
Solution:
1 14
1
n n
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