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Convergence and Divergence Tests for Series: A Comprehensive Guide, Cheat Sheet of Calculus

Columbia University in the City of New York Calculus

Cheat sheet about calculus series tests

Typology: Cheat Sheet

2018/2019

Uploaded on 09/02/2019

ekani 🇺🇸

4.7

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1. Convergence and Divergence Tests for Series

Test When to Use Conclusions

Divergence Test for any series

∞

X

n=0

anDiverges if lim

n→∞

|an| 6= 0.

Integral Test

∞

X

n=0

anwith an≥0 and andecreasing Z∞

1

f(x)dx and

∞

X

n=0

anboth converge/diverge

where f(n) = an.

Comparison Test

∞

X

n=0

anand

∞

X

n=0

bn

∞

X

n=0

bnconverges =⇒

∞

X

n=0

anconverges.

if 0 ≤an≤bn

∞

X

n=0

andiverges =⇒

∞

X

n=0

bndiverges.

Limiting Comparison Test

∞

X

n=0

an,(an>0). Choose

∞

X

n=0

bn, (bn>0)

if lim

n→∞

an

bn

=Lwith 0 <L<∞

∞

X

n=0

anand

∞

X

n=0

bnboth converge/diverge

if lim

n→∞

an

bn

= 0

∞

X

n=0

bnconverges =⇒

∞

X

n=0

anconverges.

if lim

n→∞

an

bn

=∞

∞

X

n=0

bndiverges =⇒

∞

X

n=0

andiverges.

Convergent test

∞

X

n=0

(−1)nan(an>0) converges if

for alternating Series lim

n→∞

an= 0 and anis decreasing

Absolute Convergence for any series

∞

X

n=0

anIf

∞

X

n=0

|an|converges, then

∞

X

n=0

anconverges,

(definition of absolutely convergent series.)

Conditional Convergence for any series

∞

X

n=0

anif

∞

X

n=0

|an|diverges but

∞

X

n=0

anconverges.

∞

X

n=0

anconditionally converges

For any series

∞

X

n=0

an, there are 3 cases:

Ratio Test: Calculate lim

n→∞ 



an+1

an



=Lif L < 1, then

∞

X

n=0

|an|converges ;

Root Test: Calculate lim

n→∞

n

p|an|=Lif L > 1, then

∞

X

n=0

|an|diverges;

if L= 1, no conclusion can be made.

2. Important Series to Remember

Series How do they look Conclusions

Geometric Series

∞

X

n=0

arnConverges to a

1−rif |r|<1 and diverges if |r| ≥ 1

p-series

∞

X

n=1

1

npConverges if p > 1 and diverges if p≤1

Partial preview of the text

Download Convergence and Divergence Tests for Series: A Comprehensive Guide and more Cheat Sheet Calculus in PDF only on Docsity!

Convergence and Divergence Tests for Series

Test When to Use Conclusions

Divergence Test for any series^ ∑ n^ ∞=0 an Diverges if nlim→∞ |an| 6 = 0.

Integral Test^ ∑ n^ ∞=0 an with an ≥ 0 and an decreasing^ ∫^1 ∞ f (x)dx and^ ∑ n^ ∞=0 an both converge/diverge

where f (n) = an.

Comparison Test^ ∑ n^ ∞=0 an and^ ∑ n^ ∞=0 bn^ ∑ n^ ∞=0 bn converges =⇒^ ∑ n^ ∞=0 an converges.

if 0 ≤ an ≤ bn ∑ n^ ∞=0 an diverges =⇒∑ n^ ∞=0 bn diverges.

Limiting Comparison Test^ ∑ n^ ∞=0 an, (an > 0). Choose^ ∑ n^ ∞=0 bn, (bn > 0)

if nlim→∞^ a bnn = L with 0 < L < ∞^ ∑ n^ ∞=0 an and^ ∑ n^ ∞=0 bn both converge/diverge

if nlim→∞^ a bnn = 0^ ∑ n^ ∞=0 bn converges =⇒^ ∑ n^ ∞=0 an converges.

if nlim→∞^ a bnn = ∞ ∑ n^ ∞=0 bn diverges =⇒∑ n^ ∞=0 an diverges.

Convergent test ∑ n^ ∞=0 (−1)nan (an > 0) converges if

for alternating Series (^) nlim→∞ an = 0 and an is decreasing

Absolute Convergence for any series∑ n^ ∞=0 an If∑ n^ ∞=0 |an| converges, then∑ n^ ∞=0 an converges,

(definition of absolutely convergent series.)

Conditional Convergence for any series∑ n^ ∞=0 an if∑ n^ ∞=0 |an| diverges but∑ n^ ∞=0 an converges.

∑^ ∞

n=0^ an^ conditionally converges

For any series∑ n^ ∞=0 an, there are 3 cases:

Ratio Test: Calculate nlim→∞^ ∣∣∣ an a+1n^ ∣∣∣ = L if L < 1, then∑ n^ ∞=0 |an| converges ;

Root Test: Calculate nlim→∞^ √^ n|an| = L if L > 1, then^ ∑ n^ ∞=0 |an| diverges;

if L = 1, no conclusion can be made.

Important Series to Remember Series How do they look Conclusions

Convergence and Divergence Tests for Series: A Comprehensive Guide, Cheat Sheet of Calculus

Related documents

Partial preview of the text

Download Convergence and Divergence Tests for Series: A Comprehensive Guide and more Cheat Sheet Calculus in PDF only on Docsity!

Divergence Test for any series^ ∑ n^ ∞=0 an Diverges if nlim→∞ |an| 6 = 0.

Integral Test^ ∑ n^ ∞=0 an with an ≥ 0 and an decreasing^ ∫^1 ∞ f (x)dx and^ ∑ n^ ∞=0 an both converge/diverge

Comparison Test^ ∑ n^ ∞=0 an and^ ∑ n^ ∞=0 bn^ ∑ n^ ∞=0 bn converges =⇒^ ∑ n^ ∞=0 an converges.

if 0 ≤ an ≤ bn ∑ n^ ∞=0 an diverges =⇒∑ n^ ∞=0 bn diverges.

Limiting Comparison Test^ ∑ n^ ∞=0 an, (an > 0). Choose^ ∑ n^ ∞=0 bn, (bn > 0)

if nlim→∞^ a bnn = L with 0 < L < ∞^ ∑ n^ ∞=0 an and^ ∑ n^ ∞=0 bn both converge/diverge

if nlim→∞^ a bnn = 0^ ∑ n^ ∞=0 bn converges =⇒^ ∑ n^ ∞=0 an converges.

if nlim→∞^ a bnn = ∞ ∑ n^ ∞=0 bn diverges =⇒∑ n^ ∞=0 an diverges.

Convergent test ∑ n^ ∞=0 (−1)nan (an > 0) converges if

Absolute Convergence for any series∑ n^ ∞=0 an If∑ n^ ∞=0 |an| converges, then∑ n^ ∞=0 an converges,

Conditional Convergence for any series∑ n^ ∞=0 an if∑ n^ ∞=0 |an| diverges but∑ n^ ∞=0 an converges.

∑^ ∞

For any series∑ n^ ∞=0 an, there are 3 cases:

Ratio Test: Calculate nlim→∞^ ∣∣∣ an a+1n^ ∣∣∣ = L if L < 1, then∑ n^ ∞=0 |an| converges ;

Root Test: Calculate nlim→∞^ √^ n|an| = L if L > 1, then^ ∑ n^ ∞=0 |an| diverges;

Geometric Series^ ∑ n^ ∞=0 arn^ Converges to 1 −a r if |r| < 1 and diverges if |r| ≥ 1

p-series ∑ n^ ∞=1 n^1 p Converges if p > 1 and diverges if p ≤ 1