Calculus Of One Real Variable By Pheng Kim Ving

11.3 
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1. Tests For Convergence 
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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2. The pIntegrals 
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 pintegrals.
Fig. 2.1

Fig. 2.2

Fig. 2.3

Fig. 2.4

Fig. 2.5

Fig. 2.6
Graphs of y = 1/x^{p}, x > 0, p > 0.

Fig. 2.7
y = 1/(x a)^{p}, where p > 0, is continuous on (a, b].

Fig. 2.8
y = 1/(b x)^{p}, where p > 0, is continuous on [a, b).

The pIntegrals
All of the 4 integrals above with exponent p at the denominators are called the pintegrals. 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
pintegrals are basictype improper integrals.
Each integral above is called a pintegral. Note that part ii is a special case of the 1st integral of part iii where a = 0. 
Proof
Therefore, similarly to part ii:
For each of the following integrals, determine whether it converges or diverges, without actually calculating it.
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 pintegral, then we may draw
conclusion about the integral of f .
Fig. 3.1

Fig. 3.2

Fig. 3.3

Fig. 3.4

The proof of Theorem 3.1 below makes use of the following property of the real numbers.
Fundamental Property Of The Real Numbers
Fig. 3.5

Fig. 3.6

Theorem 3.1 The
Standard Comparison Test (SCT)
This test for convergence of a basictype improper
integral is called the standard comparison test, abbreviated as 
Proof
EOP
Remarks 3.1
Establish the convergence or divergence of the following integral without actually calculating it.
converges.
EOS
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4. The Limit Comparison Test (LCT) 
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 basictype 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.
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
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
NonpIntegrals
The pintegrals
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.
Problems & Solutions 
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:
2. Establish the convergence or divergence of each of the following integrals without actually calculating it.
Solution
3. For each of the following integrals, decide whether it converges or diverges, without actually computing its value.
Solution
diverges.
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 pintegral.
See Part 5.
5. Prove that:
converges without trying to compute its value.
Solution
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