CHAPTER 9-H STRATEGIES FOR TESTING SERIES

StrategiesClassify the series to determine which test to use.

1. If then the series diverges.

This is the nth Term Test for divergence.

2. Look to see if the series is a p-series . If so it will converge if p > 1 and diverge if p ≤ 1.

3. Look to see if the series is a geometric series.

If so it will converge if r < 1. If r ≥ 1 then it will diverge. ( you may need to manipulate the series algebraically to get it into the proper form)

lim 0nna

pn

1

1-nn aror ar

Strategies cont.

4. If your series is similar to a p-series or a geometric series try a comparison test. Applies

only to series with positive terms

5. If the terms of your series alternate between positive and negative

try the alternating series test.

nnn a1-or a1 n

1

Strategies cont.

6. If your series contains factorials or a constant raised to the power n try the ratio test (but don’t bother with the ratio test for a p-series).7. If the series is of the form Ʃ(bn)n try the

root test.

8. If an = f(n) where is easily

evaluated try the integral test.

9. Telescoping series: write out series and cancel terms

1

f x dx

1

1

2 1

Can best be evaluated by

1. direct comparison test

2. limit comparison test

3. test for divergence

4. hoping for a fire drill right now

n

n

n

1.

1

1

3

1 8

Can best be evaluated using

1. root test

2. ratio test

3. integral test

4. alternating series test

n n

nn

n

n

2.

2 cont. Find the limit

21

125

Can best be evaluated using

1. ratio test

2. root test

3. integral test

4. alternating series test

n

n

n

n

3.

2

31

1

1

Can best be evaluated using

1. direct comparison test

2. limit comparison test

3. ratio test

4. ask the teacher

n

n

n

4.

3

2

Using the limit comparison test we can

compare the terms in our series with

11.

12.

13.

14.

n

n

n

n

4. cont.

2

3

11The lim is

1

1. 1

2. 0

3.

4. DNE

n

nn

n

4. cont

1

21

Can best be evaluated by

1. direct comparison test

2. integral test

3. root test

4. test for divergence

n

n

e

n

5.

1

21

The integral can be evaluated using

1. integration by parts

2. trigonometric substitution

3. partial fraction decomposition

4. substitution

xedx

x

5. cont

The result of the integration is

1. 1 converges

2. 1 converges

3. diverges

4. 0 diverges

e

e

5. cont

1

2

!

Can best be evaluated using

1. ratio test

2. test for divergence

3. limit comparison test

4. direct comparison test

n

n n

6.

1The ratio of is

1. 2

22.

23.

1

14.

2

n

n

a

a

k

k

k

k

6. cont

2 1

1

3

Can best be evaluated by

1. test for divergence

2. limit comparison test

3. ratio test

4. root test

n

nn n

7.

8. Determine the convergence or divergence of the series

1 12

1

n n

n

9. Determine the convergence or divergence of the series

123

3

243

1

n nn

n

10. Determine the convergence or divergence of the series

1

2

n

nne

11. Determine the convergence or divergence of the series

14

3

11

n

n

n

n

12. Determine the convergence or divergence of the series

1 32

1

nn

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