MAT1228 Classwork
MAT 1236 11.3 Integral Test Handout
Interesting Property about Convergence
Finite number of terms does not change the convergence of a series
[pic]
The Integral Test
If [pic]is continuous, positive, decreasing on [pic], and [pic],
then both [pic], [pic] converge/diverge.
• Note that in this case, [pic] for all [pic].
Remarks
• Also true for [pic]and [pic]
• “decreasing” can be replaced by “eventually decreasing”
• Usually it is obvious that [pic]is continuous, positive on [pic]. Typically, all you need to show is that [pic] is decreasing on [pic].
Why?
| |[pic] |
|[pic] | |
| |[pic] |
|[pic] | |
Example 1 [pic]
Step1: Declare the function [pic].
Let [pic]
Step 2: Show that the function is decreasing on [pic].
|[pic] |
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|[pic] on [pic]. So, [pic]is decreasing on[pic]. |
Step 3: State that [pic]satisfy the hypothesis of the integral test.
|Therefore, [pic]is continuous, positive and decreasing on [pic] and [pic]. |
Step 4: Find the convergence of the improper integral.
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Step 5: Make the conclusion.
By the integral test, [pic]is also .
|p-series |Classwork |
| |1. Determine whether the series is convergent or divergent. |
| |[pic] |
| |Step1: Declare the function [pic]. |
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| |Let [pic] |
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|Example 2 [pic] | |
| |Step 2: Show that the function is decreasing on [pic]. |
| |[pic] |
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| |[pic] on [pic]. So, [pic]is decreasing on[pic]. |
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| |Step 3: State that [pic]satisfy the hypothesis of the integral|
| |test. |
|[pic] is convergent. (p-series, [pic]) |Therefore, [pic]is continuous, positive and decreasing on |
| |[pic] and [pic]. |
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| |Step 4: Find the convergence of the improper integral |
|Example 3 [pic] | |
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|[pic] | |
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|[pic] is convergent. (p-series, [pic]) | |
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|[pic] is convergent. (p-series, [pic]) | |
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|[pic] is also convergent. | |
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| |Step 5: Make the conclusion. |
| |By the integral test, [pic]is also |
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Integral Test: If [pic]is continuous, positive, decreasing on [pic], and [pic], then both [pic], [pic] converge/diverge.
Reminder
Improper integrals are defined by limits.
[pic]
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