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Radius of convergence - PowerPoint PPT Presentation

Section 9.4b. Radius of convergence. Recall the Direct Comparison Test from last class…. To apply this test, the terms of the unknown series must be nonnegative . This doesn’t limit the usefulness of this test, because we can apply it to the absolute value of the series.

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Presentation Transcript
Section 9.4b

Recall the Direct Comparison Test from last class…

To apply this test, the terms of the unknown series must be

nonnegative. This doesn’t limit the usefulness of this test,

because we can apply it to the absolute value of the series.

Definition: Absolute Convergence

If the series of absolute values converges, then

converges absolutely.

Thm: Absolute Convergence Implies Convergence

If converges, then converges.

A quick example with this new rule…

Show that the given series converges for all x.

This series

has no negative terms, and it is term-by-term less than or equal to:

Which we know

converges to e.

Therefore, converges by direct comparison.

Since converges absolutely, it converges.

Also recall this theorem from last class…

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 elsewhere

The number R is the radius of convergence

Using the Ratio Test to find radius of convergence

Find the radius of convergence of the given series.

Check for absolute convergence using the Ratio Test:

Using the Ratio Test to find radius of convergence

Find the radius of convergence of the given series.

So the ratio of each term to the previous term:

Which we need to be less than one:

The series converges absolutely (and hence converges) on

this interval, and diverges when x < –10 and for x > 10.

The radius of convergence is 10

Practice Problems

Find the radius of convergence of the given power series.

This is a geometric series with

So it will only converge when

The radius of convergence is 1

Practice Problems

Find the radius of convergence of the given power series.

Ratio Test for absolute convergence:

or

The series converges for

The radius of convergence is 1/3

Practice Problems

Find the radius of convergence of the given power series.

Ratio Test for absolute convergence:

The series converges for all values of x.

8

Practice Problems

Find the radius of convergence of the given power series.

Ratio Test for absolute convergence:

Practice Problems

Find the radius of convergence of the given power series.

Ratio Test for absolute convergence:

or

The series converges for

The radius of convergence is 4

Practice Problems

Find the radius of convergence of the given power series.

Ratio Test for absolute convergence:

The series only converges for x = 4.

The radius of convergence is 0

Practice Problems

Find the interval of convergence of the given series and,

within this interval, the sum of the series as a function of x.

This is a geometric series with:

It converges when:

Interval of convergence:

Practice Problems

Find the interval of convergence of the given series and,

within this interval, the sum of the series as a function of x.

This is a geometric series with:

Sum

Practice Problems

Find the interval of convergence of the given series and,

within this interval, the sum of the series as a function of x.

This is a geometric series with:

It converges when:

Which is always!!!

Interval of convergence:

Sum