11.3 Tests For Convergence Of Improper Integrals
| 11.3 |
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There are improper integrals that can’t be evaluated by the fundamental theorem of calculus because the antiderivatives of their integrands can’t be found. In this situation, we may still be able to determine whether they converge or not by testing their convergence, which is done by comparing them to simpler improper integrals whose behavior (convergence
or divergence) is known.
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The tests for convergence of improper integrals are done by comparing these integrals to known simpler improper
integrals. We are now going to examine some of such integrals. They’re known as the p-integrals.
| Fig. 2.3
|
The p–Integrals
All of the 4 integrals above with exponent p at the denominators are called the p-integrals. To distinguish between them
we specify what their improper point is. Their basic terminology is summarized in the table below.
Observe that the “at ” in the name of an integral is used to specify the improper point of the integral. Remark that the
p-integrals are basic–type improper integrals.
Theorem 2.1 – The p–Integrals
| Each integral above is called a p-integral. Note that part ii is a special case of the 1st integral of part iii where a = 0. |
Proof
Therefore, similarly to part ii:
Example 2.1
For each of the following integrals, determine whether it converges or diverges, without actually calculating it.
Solution
EOS
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| 3. The Standard Comparison Test (SCT) |
Suppose we have a function f and we want to know if its integral converges or diverges. If f(x) can be compared to the
integrand of a p-integral, then we may draw conclusion about the integral of f .
The proof of Theorem 3.1 below makes use of the following property of the real numbers.
Fundamental Property Of The Real Numbers
Theorem 3.1 – The Standard Comparison Test (SCT)
| This test for convergence of a basic-type improper integral is called the standard comparison test, abbreviated as |
Proof
EOP
Remarks 3.1
Example 3.1
Establish the convergence or divergence of the following integral without actually calculating it.
Solution
converges.
EOS
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| 4. The Limit Comparison Test (LCT) |
The SCT Doesn’t Always Apply
The Ratio Of The Integrands
Theorem 4.1 – The Limit Comparison Test (LCT)
| Suppose: where L is some finite positive number. Then the improper integrals of f and g with the same limits of integration behave the same way, ie either both converge or both diverge. Since this test for convergence of a basic-type improper integral makes use of a limit, it’s called the limit comparison test, abbreviated as LCT. |
Proof
EOP
When we can’t find an improper integral to be used to apply the SCT to a given improper integral, we’ll try the LCT.
Example 4.1
Determine whether the following integral converges or diverges without calculating it:
Solution 1
Thus, by the LCT, the given integral converges.
EOS
Solution 2
EOS
Example 4.2
Establish the convergence or divergence of this integral without actually calculating it:
Solution
EOS
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Comparisons Between Proper Integrals
Comparisons between proper integrals derive from the properties of definite integrals, and we are already aware of them.
See Section 9.3 Theorem 6.1 Part 6. Recall that all proper integrals are finite numbers, therefore they all are convergent.
However, we may want to compare the proper integral of a function f to another proper integral if an antiderivative of f
can’t be found. For an example see Problem & Solution 4.
Comparisons With Non-p-Integrals
The p-integrals are not the only integrals used in comparison tests. There are other functions that sometimes have to be
used. For an example illustration see Problem & Solution 4.
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1. For each of the following integrals, determine whether it converges or diverges without actually calculating it.
Solution
a. The only improper point is 0. The breaking is:
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2. Establish the convergence or divergence of each of the following integrals without actually calculating it.
Solution
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3. For each of the following integrals, decide whether it converges or diverges, without actually computing its value.
Solution
diverges.
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Solution
b. We have:
Note
In part b, the first comparison is between proper integrals, and the second is made to an integral that isn’t a p-integral.
See Part 5.
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5. Prove that:
converges without trying to compute its value.
Solution
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