Harold’s Series Convergence Tests Cheat Sheet

Harold's Series Convergence Tests Cheat Sheet

24 March 2016

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Divergence or nth Term Test Geometric Series Test

p - Series Test

Series: =1

Series: =0

Condition(s) of Convergence: None. This test cannot be used to show convergence.

Condition(s) of Divergence:

lim

0

4

Alternating Series Test

Condition of Convergence: || < 1

Sum: = lim (1-) =

1-

1-

Condition of Divergence: || 1

5

Integral Test

Series:

=1

1

Condition of Convergence: > 1

Condition of Divergence: 1

6

Ratio Test

Series: =1 (-1)+1

Condition of Convergence:

0 < +1

lim

=

0

or if =0 || is convergent

Condition of Divergence: None. This test cannot be used to show divergence.

* Remainder: || +1

7

Root Test

Series: =1

Condition of Convergence:

lim

||

<

1

Condition of Divergence:

lim

||

>

1

* Test inconclusive if

lim

||

=

1

10

Telescoping Series Test

Series: =1 (+1 - )

Condition of Convergence:

lim

=

Condition of Divergence: None

Series: =1 when = () 0

and () is continuous, positive and decreasing

Condition of Convergence: 1 () converges

Condition of Divergence: 1 () diverges

* Remainder: 0 < () 8

Direct Comparison Test

(, > 0)

Series: =1

Condition of Convergence:

lim

|+ 1 |

<

1

Condition of Divergence:

lim

|+ 1 |

>

1

* Test inconclusive if

lim

|+ 1 |

=

1

9

Limit Comparison Test

({}, {} > 0)

Series: =1

Series: =1

Condition of Convergence:

0 < and =0 is absolutely

convergent

Condition of Convergence:

lim

=

>

0

and =0 converges

Condition of Divergence: 0 <

and =0 diverges

NOTE: 1) May need to reformat with partial fraction expansion or log rules. 2) Expand first 5 terms. n=1,2,3,4,5. 3) Cancel duplicates. 4) Determine limit L by taking the limit as .

5) Sum: = 1 -

Condition of Divergence:

lim

=

>

0

and =0 diverges

NOTE: These tests prove

convergence and divergence, not

the actual limit or sum S.

Sequence:

lim

=

(, +1, +2, ...)

Series: =1 = ( + +1 + +2 + )

Copyright ? 2011-2016 by Harold Toomey, WyzAnt Tutor

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Choosing a Convergence Test for Infinite Series

Courtesy David J. Manuel

Do

the individual

No

terms approach 0?

Series Diverges by the Divergence Test

Yes

Does the series

Yes

alternate

signs?

No

Do individual terms have factorials or exponentials?

No

Use Ratio Test Yes

(Ratio of Consecutive Terms)

Is individual term easy to integrate?

Use Integral Test

Yes

No

Use Alternating Series Test (Do absolute value of terms go to 0?)

Do individual terms involve fractions with

powers of n?

No

Yes

Use Comparison Test or Limit Comp. Test (Look at dominating terms)

Copyright ? 2011-2016 by Harold Toomey, WyzAnt Tutor

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