9.4 Radius of convergence comp.notebook...Ex. 7 Radius of Convergence 0 Find the radius of...
Transcript of 9.4 Radius of convergence comp.notebook...Ex. 7 Radius of Convergence 0 Find the radius of...
9.4 Radius of convergence comp.notebook
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94 Radius of Convergence
In this section we will investigate convergence using several tests. This is important because there are different constraints that govern convergence depending on the type of power series that is under investigation.
Ex. 1 Illustrating the importance of convergence
Consider the mathematical sentence:
For what values of x is this an identity? Support your answer graphically.
Graphing helps support the idea of convergence, but it does NOT PROVE convergence or divergence.
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1. 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.
2) The series
3) The series
The number R
and the set of all values x for which the series converges is called the
The radius of convergence completely determines the interval of convergence if R is either zero or infinite.
For we still need to look at the
endpoints of the interval.
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Ex. 2
Find the radius of convergence and the interval of
convergence for
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2. nth Term Test (For Divergence)
This is the most obvious test for convergence. We look for the nth term to approach zero.
Ex.3 Determine if the series converges or diverges.
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COMPARING NONNEGATIVE SERIES
Sometimes if we do not know if a series converges or diverges, we can look at a similar, known convergent series. This is used with a series that has nonnegative numbers.
3. Direct Comparison Test
Let Ʃan be a series with NO negative terms.
(a) Ʃan converges if
(b) Ʃan diverges
Ex.4 Prove that converges for all real x.
TRY Prove that converges.
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4. ABSOLUTE CONVERGENCE
If the series Ʃ|an| of absolute values converges, then Ʃan Absolute convergence implies convergence but the reverse is not necessarily true.
Ex. 5 Using Absolute Convergence
Show that converges for all x.
TRY Show that converges for all x.
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5. RATIO TEST
Let Ʃan be a series with positive terms, and with
Then,
(i) The series converges if
(ii) The series diverges if
(iii)The test is inconclusive
Complete exploration 1 on page 508.
Ex. 6 Finding the Radius of Convergence
Find the radius and the interval of convergence for
TRY Find the radius and the interval of convergence for
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Ex. 7 Radius of Convergence 0
Find the radius of convergence of the series
TRY
Find the radius of convergence of the series
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Ex. 8 Determining convergence of a series
Determine the convergence or divergence of the
series
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6. Telescoping Series
A telescoping series is so called because the partial sums all cancel out (collapse) and we are left with the initial and the final terms the two ends of a hand held telescope.
Ex.9
Find the sum of
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ENDPOINT CONVERGENCE
We need to take into consideration endpoints when investigating convergence.
Complete Exploration 2 on page 509