Section 83b Sometimes we cannot evaluate an improper i ntegral directly In these cases we first try to d etermine whether it converges or diverges Diverges End of story were done ID: 248467
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Slide1
Tests for Convergence and Divergence
Section 8.3bSlide2
Sometimes we cannot evaluate an improperintegral directly
In these cases, we first try to
determine whether it converges or diverges.
Diverges???... End of story, we’re done.
Converges???... Use
numerical methods
to
Approximate the value of the integral.Slide3
Investigating Convergence
Does the integral converge?
Definition from last class:
We cannot directly evaluate this last integral directly because
there is no simply formula for the
antiderivative
of the integrand…
Instead, compare the integral to one we
can
evaluate…
Graph both functions in [0, 3] by [–0.5, 1.5]:
And note that for Slide4
Investigating Convergence
Does the integral converge?
Thus, for any
Because the integral of the larger function converges, the integral
o
f the smaller function converges as well. However, this does
n
ot tell us the value of the improper integral, except that it is
p
ositive and less than 0.368…Slide5
Investigating Convergence
Does the integral converge?
Evaluate numerically:
Try graphing:
in [0, 20] by [–0.1, 0.3]
What does the graph suggest as ?Slide6
Direct Comparison Test
Let and be continuous on and let
for all . Then
1. converges if converges.
2. diverges if diverges.Slide7
Investigating Convergence
Does the integral converge?
Compare this integrand to :
on
So because the integral of this “larger” function converges,
the original integral converges as well!Slide8
Investigating Convergence
Does the integral converge?
Compare this integrand to :
on
So because the integral of this “smaller” function diverges,
the original integral diverges as well!Slide9
Limit Comparison Test
If the positive functions and are continuous on
a
nd if
then
and
b
oth converge
or diverge
If two functions grow at the same rate as
x
approaches infinity (recall Section 8.2?), then their
integrals from
a
to infinity behave
a
like; they both
converge or both diverge.Slide10
Investigating Convergence
Show that converges by comparison with
Find and compare the two integral values.
The two functions are positive and continuous on the given
interval… Use the new rule:
Form
Thus, both integrals converge. Now to find their value…Slide11
Investigating Convergence
Show that converges by comparison with
Find and compare the two integral values.
(from earlier in class)Slide12
Investigating Convergence
Show that converges.
Compare with:
This integral converges (an
e
xample from earlier in class) to
a
value of about 0.368…
Thus, the original integral also converges!Slide13
Investigating Convergence
Use integration, the direct comparison test, or the limit
comparison test to determine whether the given integral
converges or diverges.
The integral convergesSlide14
Investigating Convergence
Use integration, the direct comparison test, or the limit
comparison test to determine whether the given integral
converges or diverges.
Compare to :
on
Sinc
e this integral diverges,
the given integral diverges