In the previous section, we studied positive series, but we - PowerPoint Presentation

Slide1

https://www.khanacademy.org/math/integral-calculus/sequences-series-approx-calc/convergence-divergence-tests/v/alternating-series-test

https://www.khanacademy.org/math/integral-calculus/sequences-series-approx-calc/convergence-divergence-tests/v/conditional-and-absolute-convergenceSlide2

In the previous section, we studied positive series, but we still lack the tools to analyze series with both

positive and negative terms

. One of the keys to understanding such series is the concept of

absolute convergence

.Slide3

Verify that the seriesSlide4
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may converge without

converging absolutely

. In this case, we say that

is

conditionally convergent

.

DEFINITION

Conditional Convergence

An infinite series

converges conditionally

if

converges

but

diverges

.

The series in the previous example does not converge absolutely, but we still do not know whether or not it converges. A seriesSlide6

An alternating series with

terms decreasing in magnitude

. The sum is the signed area, which is less than

a

1.

Alternating Series

where the terms

a

n

are

positive and decrease to zero

.Slide7

THEOREM 2

Leibniz Test for Alternating Series

Assume that

{

a

n

}

is a

positive

sequence

that is

decreasing

and

converges to 0

:

Then the following alternating series converges:

Furthermore,

Example

Next ExampleSlide8
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diverges. Therefore, S is conditionally convergent but not

absolutely convergent.

Therefore,

S

converges by the Leibniz Test

. Furthermore, 0 < S < 1. However, the positive series

Show that

converges conditionally

and that

0 <

S

< 1

.

The terms

are positive and decreasing, and

(A) Partial sums of

S

=

(B) Partial sums Slide10

c

cSlide11

Alternating Harmonic Series

Show that

converges conditionally. Then:

(a)

Show that(b) Find an

N such that SN approximates S with an error less than 10−3.

c

c

c

c

cSlide12

Alternating Harmonic Series

Show that

converges conditionally. Then:

(a)

Show that

(b) Find an N such that SN

approximates S with an error less than 10−3.

We can make the error less than 10

−3

by choosing

N

so thatSlide13

CONCEPTUAL INSIGHT

The convergence of an infinite series

depends on two factors:

(1)

how quickly

a

n

tends to zero, and

(2)

how much cancellation takes place among the terms. ConsiderSlide14
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