MAT 1236Calculus III

Section 11.4

The Comparison Tests

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HW ….

WebAssign 11.4 Write out your solutions carefully Final is coming up!

Reminder to Wai

Do the first 3 examples on the board without erasing

Overwrite on the first 3 examples to save time

Preview

Comparison Test Limit Comparison Test Only work for series with positive terms Compare with standard series The nature of the comparison is term-by-term

Examples

12 729

7

n nn

1

ln

n n

n

22 1

1

n n

Comp. Test Limit Comp. Test

Examples

12 729

7

n nn

1

ln

n n

n

22 1

1

n n

Comp. Test Limit Comp. Test

How does it diverge?

1

0 for all 1, 2,3,

n

nn

a n

a

The Comparison Test

Suppose for all .

If is convergent then is also convergent

If is divergent then is also divergent

Be Careful!!!!!

Suppose for all .

If is convergent then is also convergent

However,

If is convergent then there is no conclusion for

Example 1

12 729

7

n nn

Series to compare with

Geometric Series Harmonic Series -series

1

1

1

1

n

n

n

pn

ar

k

n

k

n

Remarks

For convenience, we will call the following series a p-series.

It has the same convergence as

n mp

k

n

1p

n

k

n

Series to compare with

Geometric Series Harmonic Series -series

2 2

7 7 VS

9 2 7 9n n n

12 729

7

n nn

Example 1

Write down the general terms of the two series Write down the comparison and range Find the convergence of the comparison series Make the conclusion by quoting the name of the

comparison test

Common Mistake/Misconception

Comparing the series instead of comparing the terms

1

21

2 9

7

729

7

nn nnn

The series are not comparable unless you first show that they are both convergent.

The comparison test is based on the comparison of general terms.

STOP

Examples

12 729

7

n nn

1

ln

n n

n

22 1

1

n n

Comp. Test Limit Comp. Test

Example 2

1

ln

n n

n

1

1 1 1

1 1 n

pn n n

arn n

Remark: Need to justify the inequality if not obvious.

Examples

12 729

7

n nn

1

ln

n n

n

22 1

1

n n

Comp. Test Limit Comp. Test

Example 3

22 1

1

n n

1

1 1 1

1 1 n

pn n n

arn n

Examples

12 729

7

n nn

1

ln

n n

n

22 1

1

n n

Comp. Test Limit Comp. Test

?

The Limit Comparison Test (L.C.T.)

Suppose

If

then both , converge or diverge

no. finite a is where0lim ccb

a

n

n

n

The Limit Comparison Test (L.C.T.)

Suppose

If

then both , converge or diverge

no. finite a is where0lim ccb

a

n

n

n

Example 4

12 729

7

n nn

1

1 1 1

1 1 n

pn n n

arn n

Examples

12 729

7

n nn

1

ln

n n

n

22 1

1

n n

Comp. Test Limit Comp. Test

?

Example 5

1

ln

n n

n

1

1 1 1

1 1 n

pn n n

arn n

Examples

12 729

7

n nn

1

ln

n n

n

22 1

1

n n

Comp. Test Limit Comp. Test

??

Example 6

22 1

1

n n

Examples

12 729

7

n nn

1

ln

n n

n

22 1

1

n n

Comp. Test Limit Comp. Test

??