Chapter 9.6Chapter 9.6Chapter 9.6Chapter 9.6

THE RATIO AND ROOT TESTSTHE RATIO AND ROOT TESTS

After you finish your HOMEWORK you will be able to…

• Use the Ratio Test to determine whether a series converges or diverges

• Use the Root Test to determine whether a series converges or diverges.

• Know when to use what test!

THEOREM 9.17RATIO TEST

Condition: Let be a series with nonzero terms.

1. converges absolutely if

2. diverges if or

3. The Ratio Test is inconclusive if

1lim 1.n

nn

a

a

na

na

1lim 1n

nn

a

a

1lim .n

nn

a

a

1lim 1.n

nn

a

a

na

USING THE RATIO TEST

Determine the convergence or divergence of

Does this series have nonzero terms?

1

1

3

n

nn

n

You got it! All terms are nonzero…that’s a pretty easy condition to fulfill, don’t you think?

Now we need to find and na

1

1 1

1 1 1 and

3 3

n n

n nn n

n na a

1.na

So let’s make a ratio!

1

11

1

1

1 1

31

3

1 1

31 1

3

n

nn

nn

n

n n

n n

na

a n

n

nn

n

Now for the test…

Conclusion:By the ratio test,

this is an absolutely convergent series, and therefore convergent.

11 1

lim lim3

1lim

3

1

lim3

11lim

3

11

3

n

n nn

n

n

n

na

a n

n

n

nn

nn

n

THEOREM 9.18ROOT TEST

Condition: Let be a series.

1. converges absolutely if

2. diverges if or

3. The Root Test is inconclusive if

1

lim 1.nnna

na

na

na 1

lim 1nn

na

1

lim .nn

na

1

lim 1.nn

na

USING THE ROOT TEST

Determine the convergence or divergence of

Is this a series?

1

1n

n e

Sure is! I like this condition best of all!

Now we need to find and na

11 1

and .nn nna a

e e

1

.nna

Now for the test…

Conclusion:By the root test, this

is an absolutely convergent series, and therefore convergent.

1

1 1lim lim

1lim

11

2.718...

nn

n nn n

n

ae

e

WHICH TEST TO WHICH TEST TO APPLY WHEN?!APPLY WHEN?!

WHICH TEST TO WHICH TEST TO APPLY WHEN?!APPLY WHEN?!

Shhh…Let’s talk about the Shhh…Let’s talk about the series secretsseries secrets

STEP 1Apply the nth Term Test. If , the infinite series

diverges and you’re all done!

lim 0nna

STEP 2Check if the series is a geometric

series or a p-series.• If it’s a geometric series with

then it converges.

• If it is a p-series with then the series converges.

0 1,r

1,p

STEP 3If is a positive term series, use

one of the following tests. 1. Direct Comparison Test2. Limit Comparison TestFor these tests we usually compare

with a geometric or p-series.

1n

n

a

STEP 3 CONT.3. Ratio Test: When there is an n!,

or a .4. Root Test: When there is an

or some function of n to the nth power.

5. Integral Test: Must have terms that are decreasing toward zero and that can be integrated.

ncnn

na

STEP 4If is an alternating series, use

one of the following tests.1. Alternating Series Test.2. A test from step 3 applied to

. If so does .

1n

n

a

1n

n

a

1n

n

a

1n

n

a