Section 8.3: The Integral and Comparison Tests; Estimating Sums Practice HW from Stewart Textbook...
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Transcript of Section 8.3: The Integral and Comparison Tests; Estimating Sums Practice HW from Stewart Textbook...
Section 8.3: The Integral and Comparison Tests; Estimating Sums
Practice HW from Stewart Textbook (not to hand in)
p. 585 # 3, 6-12, 13-25 odd
In this section, we want to determine other methods for determining whether a series converges or diverges.
The Integral TestFor a function f, if f (x) > 0, is continuous and decreasing for and , then
either both converge or both diverge.
Mx )(nfan
MMnn dxxfa )( and
Note
The integral test is only a test for convergence or divergence. In the case of convergence, it does not find a value for the sum of the series.
Example 1: Determine the convergence or divergence of the series
Solution:
1 141
n n
Example 2: Determine the convergence or divergence of the series
Solution: (In typewritten notes)
22)(ln
1 n nn
Example 3: Show why the integral test cannot be used to analyze the convergence or divergence of the series
Solution:
1
1
)1(
n
n
n
p-Series and Harmonic SeriesA p-series series is given by
If p = 1, then
is called a harmonic series.
1 41
31
21
111
npppppn
1 41
31
2111
n n
Convergence of p – series
A p-series
1. Converges if p > 1. 2. Diverges if .
1 41
31
21
111
npppppn
1p
Example 4: Determine whether the p-series
is convergent or divergent.
Solution:
251
161
91
411
Example 5: Determine whether the p-series is convergent or divergent.
Solution:
1
1
n n
Example 6: Determine whether the p-series is convergent or divergent.
Solution:
1
1
n n
Making Comparisons between Series that are Similar
Many times we can determine the convergence or divergence of a series by comparing it with the known convergence or divergence of a related series. For example,
is close to the p-series ,
is close to the geometric series .
12 231
n n
12
1
n n
1 134
nn
n
1 34
nn
n
Under the proper conditions, we can use a series where it is easy to determine the convergence or divergence and use it to determine convergence or divergence of a similar series using types of comparison tests. We will examine two of these tests – the direct comparison test and the limit comparison test.
Direct Comparison Test Suppose that and are series only with positive terms ( and ).
1. If is convergent and for all terms n, is convergent.
2. If is divergent and for all terms n, is divergent.
na
na
na nb
nb
nb
0na 0nb
nn ba
nn ba
Example 7: Determine whether the series is convergent or divergent.
Solution:
12 231
n n
Example 8: Determine whether the series is convergent or divergent.
Solution:
1 134
nn
n
Example 9: Demonstrate why the direct comparison test cannot be used to analyze the convergence or divergence of the series
Solution:
02 11)1(
n
n
n
Limit Comparison Test
Suppose that and are series only with positive terms ( and ) and
where L is a finite number and L > 0.
Then either and either both converge or and both diverge.
na nb
0na 0nb
Lba
n
nn
lim
na nb
nb na
Note: This test is useful when comparing with a p-series. To get the p-series to compare with take the highest power of the numerator and simplify.
Example 10: Determine whether the series is convergent or divergent.
Solution:
12 52
35
n nnn