Comparison Tests

is a geometric series.

is a p-series.

is easily integrated.

0 2

1

nn

0 2nn

n

02

1

n n

02 1

1

n n

22 3

n

nan

22

2

3

n

nan

To check the convergence of an unknown series, compare it to a series you know.

is not

is not

is not

Direct Comparison Test (DCT)

For series with positive terms, let 0 ≤ an ≤ bn for all n

If converges, then (something smaller) converges

If diverges, then (something larger) diverges

na

na nb

nb

Examples

Does converge or diverge?

1 23

1

nn

Examples

Does converge or diverge?

1 2

ln5

n n

n

Try:

12 23

1

n n

1 54

3

nn

n

1 12

1

nn

Limit Comparison Test (LCT) na nbGiven two series and . Let’s compare the terms

of each using limits.

If , then . 1lim

n

n

n b

ann ba

So if bn converges, an also converges. If bn diverges, an

also diverges.

cb

a

n

n

n

limIf , then .

nn cba So if bn converges, an also converges. If bn diverges, an

also diverges.

The constant doesn’t change the convergence of the series since it can be factored out.

Limit Comparison Test (LCT) na nbGiven two series and . Let’s compare the terms

of each using limits.

If , then . 0lim

n

n

n b

ann ba

So if bn converges, the smaller an also converges. If bn diverges, we don’t know about an .

n

n

n b

alimIf , then .

nn ba If bn diverges, the larger an

also diverges. If bn

converges, we don’t know about an. We are back to the Direct Comparison Test.

Limit Comparison Test (LCT)For series with positive terms,

If a finite, nonzero constant, then the series

behave the same, both diverge or both converge.

n

n

n b

alim

So what about ?

1 12

1

nn

More examples: Converge or Diverge.

12 543

1

n nn

1 23

1

n n

More examples: Converge or Diverge.

135

2

4

10

n nn

n

1 !

1

n n

Practice:p.573 #1-25odd, 27-34