SECT. 9-D COMPARISON TESTS

The Direct comparison Test If 0 < an < bn for all n and positive terms, then:

If the larger series converges, then the smaller series must also converge.

If the smaller series diverges then the larger series must also diverge

converges a then converges, if1 1n

n

nnb

diverges b then diverges, if1 1n

n

nna

The Direct comparison Test Given a seriesCompare with a similar Geometric or p- Series

When choosing a series for comparison you can disregard all but the highest power in the numerator and denominator

When choosing an appropriate p-series, you must choose one with the same nth term

1 1n2

n

3 2 with 14

2 comparen

n

nnn

1. Does converge or diverge?

12 342

5n nn

12

12 2

5 ? 342

5nn nnn

2. Does converge or diverge?

1 231

nn

11 21 ?

231

nn

nn

3. Does converge or diverge?

122

1n n

12

12

1 ? 44

1nn nnn

4. Does converge or diverge?

1 11

n n

11

1 ? 1

1nn nn

The Limit comparison Test If an > 0, and bn > 0 for all n , then:

Where L is finite and positive, then both converge or they both diverge

Works well when comparing a messy algebraic series to a p-series or geometric series

Lba

n

n

n

lim

11

and n

nn

n ba

5. Does converge or diverge?

15

2

5

32n n

nn

6. Does converge or diverge?

123 54n nn

n

7. Does converge or diverge?

12

)ln(n n

n

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