Section 94a Consider the sentence For what values of x is this an identity On the left is a function with domain of all real numbers and on the right is a limit of Taylor polynomials As we have already explored in previous sections check the ID: 573370
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Slide1
Convergence Tests
Section 9.4aSlide2
Consider the sentence
For what values of
x
is this an identity?
On the left is a function with domain of all real numbers,
and on the right is a limit of Taylor polynomials…
As we have already explored in previous sections (check the
graph!), these polynomials only converge to the functionover the interval (–1, 1)…
While the graphs we have used provide compelling
visual support, they don’t actually
prove
convergence…Slide3
The Convergence Theorem for Power Series
There are three possibilities for
with respect to convergence:
1. There is a positive number
R
such that the series diverges
for but converges for . The
series may or may not converge at either of the endpoints
and .
2. The series converges for every
x
.
3. The series converges at
x
=
a
and diverges elsewhereSlide4
The Convergence Theorem for Power Series
The number
R
is the
radius of convergence, and the set ofall values of
x for which the series converges is the intervalof convergence. The radius of convergence completelydetermines the interval of convergence if
R is either zero orinfinite. For all other R values, there remains the question
of what happens at the endpoints of the interval (recall thatthe table on p.477 includes intervals of convergence thatare open, half-open, and closed). The endpoint questionwill be addressed in the final section of this chapter…Slide5
The
n
th-Term Test for Divergence
d
iverges if fails to exist or is not zero.
Essentially, a convergent series must have the
n
th term go
t
o zero as
n
approaches infinity.Slide6
The Direct Comparison Test
Let be a series with no negative terms.
We can check a series for convergence by comparing it term
by term with another known convergent or divergent series.
(a) converges if there is a convergent series
with for all , for some integer .
(b) diverges if there is a divergent series of
n
onnegative terms with for all , for some
integer .Slide7
The Ratio Test
Let be a series with positive terms, and with
Then,
(a) the series
converges
if ,
(b) the series
diverges
if ,
(c) the test is
inconclusive
if .Slide8
Guided Practice
For each of the following, determine the convergence or
d
ivergence of the series. Identify the test (or tests) you use.
There may be more than one correct way to determineconvergence or divergence of a given series.
n
th-Term Test:
Because the “final” term fails to exist,
the series diverges.Slide9
Guided Practice
For each of the following, determine the convergence or
d
ivergence of the series. Identify the test (or tests) you use.
There may be more than one correct way to determineconvergence or divergence of a given series.
Ratio Test:
Because this ratio is greater
than one, the series diverges.Slide10
Guided Practice
For each of the following, determine the convergence or
d
ivergence of the series. Identify the test (or tests) you use.
There may be more than one correct way to determineconvergence or divergence of a given series.
This is simply a geometric series with
Because , the series converges.Slide11
Guided Practice
For each of the following, determine the convergence or
d
ivergence of the series. Identify the test (or tests) you use.
There may be more than one correct way to determineconvergence or divergence of a given series.
n
th-Term Test:
The series diverges.
LetSlide12
Guided Practice
For each of the following, determine the convergence or
d
ivergence of the series. Identify the test (or tests) you use.
There may be more than one correct way to determineconvergence or divergence of a given series.
Ratio Test:
The series converges.Slide13
Guided Practice
For each of the following, determine the convergence or
d
ivergence of the series. Identify the test (or tests) you use.
There may be more than one correct way to determineconvergence or divergence of a given series.
The series diverges.
n
th-Term Test:Slide14
Guided Practice
For each of the following, determine the convergence or
d
ivergence of the series. Identify the test (or tests) you use.
There may be more than one correct way to determineconvergence or divergence of a given series.
The series diverges.
Ratio Test:Slide15
Guided Practice
For each of the following, determine the convergence or
d
ivergence of the series. Identify the test (or tests) you use.
There may be more than one correct way to determineconvergence or divergence of a given series.
Ratio Test:Slide16
Guided Practice
For each of the following, determine the convergence or
d
ivergence of the series. Identify the test (or tests) you use.
There may be more than one correct way to determineconvergence or divergence of a given series.
The series converges.