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CALCULUS 1 PROBLEMS & SOLUTIONS
6.7.2
Tests For
Convergence Of Improper Integrals
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Review
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 p-Integrals
The tests for convergence of improper integrals is done by comparing them to
known simpler improper
integrals. We are now going to examine some of such integrals. They're known as
the p-integrals.
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Fig. 1
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Fig. 2
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Fig. 3 |
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Fig. 4
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Fig. 5
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Fig. 6
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Fig. 7
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Fig. 8
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The p-Integrals
All of the four 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.
Note that the p-integrals are basic-type
improper integrals.
Theorem 1 – The p-Integrals
¾¾¾¾¾¾¾¾¾¾¾¾
¾¾¾¾¾¾¾¾¾¾¾¾
Proof
EOP
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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
.
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Fig. 9
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Fig. 10
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Fig. 11
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Fig. 12
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The proof of Theorem 2 below makes use of the following property of the real
numbers.
Fundamental Property Of The Real Numbers
If a function F(x)
is non-decreasing for all x ³ a for some number a,
then either limx®¥ F(x) = L
for
some finite number L or limx®¥ F(x) = ¥.
This property says that if F is
non-decreasing on [a, ¥), then it has only two
possibilities when
x ® ¥:
either it approaches a finite number or it approaches ¥. It
can't approach – ¥, and it can't
oscillate between two values. Examples: limx®¥ arctan x = p/2, limx®¥ x2 =
¥. On an intuitive level,
this property is obvious. On a formal level, it's a theorem whose proof is
encountered in a more
advanced calculus course or an analysis course, where the real numbers are “ constructed”.
Remarks 1
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4. The Limit Comparison Test (LCT)
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5. Other Comparisons
Comparisons Between Proper Integrals
Comparisons between proper integrals derive from the properties of definite
integrals, and we are
already aware of them. See Section
6.2.2 Theorem 2 vi. 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 which
sometimes have to be used. For an
example illustration see Problem & Solution
4.
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Problems & Solutions
1. For each of the following
integrals, determine whether it converges or diverges, without actually
calculating it.
Solution
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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
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4.
Solution
Note
In part b, the first comparison is between proper integrals, and the
second is made to an integral
which isn't a p-integral. See Review Part 5.
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Solution
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