1. Convergence and Divergence Tests for Series Test When to ...

1. Convergence and Divergence Tests for Series

Test Divergence Test Integral Test Comparison Test Limiting Comparison Test

Convergent test for alternating Series Absolute Convergence Conditional Convergence

Ratio Test: Root Test:

When to Use

for any series an

n=0

an with an 0 and an decreasing

n=0

an and bn

n=0

n=0

if 0 an bn

an, (an > 0). Choose bn, (bn > 0)

n=0

if lim an = L n bn

n=0

with 0 < L <

if lim an = 0 n bn

if lim an = n bn

(-1)nan (an > 0)

n=0

for any series an

n=0

for any series an

n=0

For any series an,

n=0

Calculate lim an+1 = L n an

Calculate lim n |an| = L

n

Conclusions

Diverges if lim |an| = 0.

n

f (x)dx and an both converge/diverge

1

n=0

where f (n) = an.

bn converges = an converges.

n=0

n=0

an diverges = bn diverges.

n=0

n=0

an and bn both converge/diverge

n=0

n=0

bn converges = an converges.

n=0

n=0

bn diverges = an diverges.

n=0

n=0

converges if

lim an = 0 and an is decreasing

n

If |an| converges, then an converges,

n=0

n=0

(definition of absolutely convergent series.)

if |an| diverges but an converges.

n=0

n=0

an conditionally converges

n=0

there are 3 cases:

if L < 1, then |an| converges ;

n=0

if L > 1, then |an| diverges;

n=0

if L = 1, no conclusion can be made.

2. Important Series to Remember

Series Geometric Series p-series

How do they look

arn

n=0

1 np

n=1

Conclusions

a

Converges to

if |r| < 1 and diverges if |r| 1

1-r

Converges if p > 1 and diverges if p 1

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download