Series convergence tests cheat sheet PDF

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Cheat Sheet for infinite series...

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Harold’s Series Convergence Tests Cheat Sheet 24 March 2016 1

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Divergence or nth Term Test

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Geometric Series Test

Series: ∑∞ 𝑛=1 𝑎𝑛

𝑛 Series: ∑∞ 𝑛=0 𝑎𝑟

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

Condition of Convergence: |𝑟| < 1

Condition(s) of Divergence: lim 𝑎𝑛 ≠ 0 𝑛→∞

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Sum: 𝐒 = lim

𝑛→∞

𝑛+1 Series: ∑∞ 𝑎𝑛 𝑛=1 (−1)

Condition of Convergence: 0 < 𝑎𝑛+1 ≤ 𝑎𝑛 lim 𝑎𝑛 = 0 𝑛→∞

or if ∑∞ 𝑛=0 |𝑎𝑛 | is convergent

Condition of Divergence: None. This test cannot be used to show divergence. * Remainder: |𝑅𝑛 | ≤ 𝑎𝑛+1 7

Root Test Series:

∑∞ 𝑛=1

𝑎𝑛

Condition of Convergence: 𝑛 lim √|𝑎𝑛 | < 1 𝑛→∞

Condition of Divergence: 𝑛 lim √|𝑎𝑛 | > 1 𝑛→∞

* Test inconclusive if 𝑛 lim √|𝑎𝑛 | = 1 𝑛→∞

Series: ∑ ∞ 𝑛=1

𝑎(1−𝑟 𝑛 ) 1−𝑟

=

𝑎

1−𝑟

Condition of Divergence: |𝑟| ≥ 1 5

Alternating Series Test

Series: ∑ ∞ 𝑛=1 𝑎𝑛 when 𝑎𝑛 = 𝑓(𝑛) ≥ 0 and 𝑓(𝑛) is continuous, positive and decreasing Condition of Convergence: ∞ ∫1 𝑓(𝑥)𝑑𝑥 converges Condition of Divergence: ∞ ∫1 𝑓(𝑥)𝑑𝑥 diverges * Remainder: 0 < 𝑅𝑁 ≤ 8

∞ ∫𝑁 𝑓(𝑥)𝑑𝑥

Direct Comparison Test (𝑎𝑛 , 𝑏𝑛 > 0)

Series:

∑∞ 𝑛=1

𝑎𝑛

Series: ∑∞ 𝑛=1 (𝑎𝑛+1 − 𝑎𝑛) Condition of Convergence: lim 𝑎𝑛 = 𝐿 𝑛→∞

Condition of Divergence: None

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𝑛𝑝

Condition of Convergence: 𝑝>1 Condition of Divergence: 𝑝≤1 6

Integral Test

Ratio Test Series: ∑ ∞ 𝑛=1 𝑎𝑛 Condition of Convergence: 𝑎𝑛+1 lim | |1 lim | 𝑛→∞ 𝑎𝑛 * Test inconclusive if 𝑎𝑛+1 |=1 lim | 𝑛→∞ 𝑎𝑛 9

Limit Comparison Test ({𝑎𝑛 }, {𝑏𝑛 } > 0)

Series:

∑∞ 𝑛=1

𝑎𝑛

Condition of Convergence: 0 < 𝑎𝑛 ≤ 𝑏𝑛 and ∑∞ 𝑛=0 𝑏𝑛 is absolutely convergent

Condition of Convergence: 𝑎𝑛 lim =𝐿>0 𝑛→∞ 𝑏𝑛 ∞ and ∑𝑛=0 𝑏𝑛 converges

Condition of Divergence: 0 < 𝑏𝑛 ≤ 𝑎𝑛 and ∑ ∞ 𝑛=0 𝑏𝑛 diverges

Condition of Divergence: 𝑎𝑛 lim =𝐿>0 𝑛→∞ 𝑏𝑛 ∞ and ∑𝑛=0 𝑏𝑛 diverges NOTE: These tests prove convergence and divergence, not the actual limit 𝐿 or sum S.

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Telescoping Series Test

p - Series Test

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 𝑛 → ∞. 5) Sum: 𝑆 = 𝑎1 − 𝐿

Copyright © 2011-2016 by Harold Toomey, WyzAnt Tutor

Sequence: lim 𝑎𝑛 = 𝐿 𝑛→∞

(𝑎𝑛 , 𝑎𝑛+1 , 𝑎𝑛+2, …)

Series: ∑∞ 𝑛=1 𝑎𝑛 = 𝐒 (𝑎𝑛 + 𝑎𝑛+1 + 𝑎𝑛+2 + ⋯ )

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Choosing a Convergence Test for Infinite Series Courtesy David J. Manuel

Do the individual terms approach 0?

No

Series Diverges by the Divergence Test

Yes Does the series alternate signs?

Do individual terms have factorials or exponentials?

Yes

No

No

Is individual term easy to integrate?

Use Integral Test Yes

No

Do individual terms involve fractions with powers of n? No

Yes

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

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

Copyright © 2011-2016 by Harold Toomey, WyzAnt Tutor

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Use Ratio Test Yes (Ratio of Consecutive Terms)...