series with positive terms: tests for convergence, pt. 1
DESCRIPTION
Series with Positive terms: tests for Convergence, Pt. 1. The comparison test, the limit comparison test, and the integral test. In this presentation, we will base all our series at 1, but similar results apply if they start at 0 or elsewhere. Comparing series. . . - PowerPoint PPT PresentationTRANSCRIPT
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Series with Positive terms: tests for
Convergence, Pt. 1The comparison test,
the limit comparison test, and the integral test.
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Comparing series. . .Consider two series ,
with for all k.1 1
and k kk k
a b
kk ba 0
In this presentation, we will base all our series at 1, but similar results apply if they start at 0 or elsewhere.
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Comparing series. . .Consider two series ,
with for all k.
How are these related in terms of convergence or divergence?
1 1
and k kk k
a b
kk ba 0
Note that:
1 1
1 2 1 2
1 2 3 1 2 3
1 2 3 4 1 2 3 4
And so on
a ba a b ba a a b b ba a a a b b b b
What does this tell us?
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Comparing series. . .Consider two series ,
with for all k.
and k ka b
kk ba 0
Note that:
1 1
1 2 1 2
1 2 3 1 2 3
1 2 3 4 1 2 3 4
And so on
a ba a b ba a a b b ba a a a b b b b
What does this tell us?
Where does the fact that the terms are non-negative come in?
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Series with positive terms. . .
1x
Since for all positive integers k. Then
0 kx
So the sequence of partial sums is . . . 1
n
n kk
s a
Non-decreasing Bounded above Geometric
1 2x x 1 2 3x x x 1 2 3 4x x x x
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Back to our previous scenario. . . Consider two series ,
with for all k.1 1
and k kk k
a b
0 k ka b
Suppose that the series converges 1
kk
b
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Suppose that the series converges
0 0 0
Note that for all positive integers ,
n n
k k kk k k
n
a b b
1k
k
b
Non-decreasing Bounded above Geometric
So the sequence of partial sums is . . . 1
n
n kk
s a
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A variant of a familiar theorem
Suppose that the sequence is non-decreasing and bounded above by a number A. That is, . . .
ks
1 1 2 1 2 3a a a a a a A
Then the series converges to some value that is smaller than or equal to A.
1k
k
a
Theorem 3 on page 553 of OZ
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Suppose that the series diverges
0 0
n n
k kk k
a b
1k
k
a
Non-decreasing Bounded below Unbounded
So the sequence of partial sums is . . . 1
n
n kk
s b
For all n we still have
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This gives us. . . The Comparison Test: Suppose we have two series , with for all positive integers k.
If converges, so does , and
If diverges, so does .
1 1
and k kk k
a b
1k
k
b
1k
k
a
1k
k
b
1k
k
a
kk ba 0
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A related test. . .
There is a test that is closely related to the comparison test, but is generally easier to apply. . . It is called the
Limit Comparison Test
This test is not in the book!
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(One case of…) The Limit Comparison Test
Limit Comparison Test: Consider two series
with , each with positive terms.
If , then
are either both convergent or both divergent.
1 1
and k kk k
a b
n
n
ba
nlim
01 1
and k kk k
a b
Why does this work?
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(Hand waving) Answer:
Because if
Then for “large” n, ak t bk. This means that “in the long run”
0 0
and have the same convergence behavior.n n
k kk k
a t b
tba
n n
n
lim
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The Integral Test
y = a(x)
Now we add some enlightening pieces to our diagram….
Suppose that we have a sequence {ak} and we associate it with a continuous function y = a(x), as we did a few days ago. . .
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The Integral Test
y = a(x)
Look at the graph. . .What do you see?
Suppose that we have a sequence {ak} and we associate it with a continuous function y = a(x), as we did a few days ago. . .
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y = a(x)
The Integral Test
1a2a
3a4a
6a5a7a
If the integral
so does the series.
11
( ) ( )k
a x dx a k
So
converges
diverges
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The Integral Test
Now look at this graph. . .What do you see?
y = a(x)
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y = a(x)
The Integral Test
1a
2a3a
4a6a5a
7a
If the integral
so does the series.
12
( ) ( )k
a k a x dx
So
converges
diverges8a
Why 2?
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The Integral TestThe Integral Test: Suppose for all x 1, the function a(x) is continuous,
positive, and decreasing. Consider the series and the integral .
If the integral converges, then so does the series.If the integral diverges, then so does the series.
1
( )k
a k
1( )a x dx
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The Integral TestThe Integral Test: Suppose for all x 1, the function a(x) is continuous,
positive, and decreasing. Consider the series and the integral .
If the integral converges, then so does the series.If the integral diverges, then so does the series.
1
( )k
a k
1( )a x dx
Where do “positive and decreasing”
come in?