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Calculus 2 Cheat Sheet on Series Divergent Convergent, Cheat Sheet of Calculus

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Uploaded on 04/27/2021

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Calculus II Series - Things to Consider
Important: This cheat sheet is not intended to be a list of guaranteed rules to follow. This intro-
duces some hints and some ideas you may consider when choosing tests for convergence or divergence when
evaluating a given series.
It is usually a good idea to try using the Test for Divergence as a first step when evaluating a series
for convergence or divergence. If we can show that:
lim
n→∞
an6= 0
Then we can say that the series diverges without having to do any extra work.
Below are some general cases in which each test may help:
P-Series Test:
The series be written in the form: P1
np
Geometric Series Test:
When the series can be written in the form: Panrn1or Panrn
Direct Comparison Test:
When the given series, anlooks like a known, or more simple, series, bn
Limit Comparison Test:
When you can see that the series looks like another convergent or divergent series, bn
But it is hard to say whether bn> anor bn< an
Root Test:
When the series can be written in the form: P(an)n
Alternating Series Test:
When the series can be written in the form: P(1)n+1anor P(1)nan
Ratio Test:
Whenever we are given something involving a factorial, e.g. n!
Whenever we are given something involving a constant raised to the nth power, e.g. Pn+5
5n
Integral Test:
If the sequence is:
continuous
positive
decreasing (we can use the First Derivative Test here)
pf2
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Calculus II Series - Things to Consider

Important: This cheat sheet is not intended to be a list of guaranteed rules to follow. This intro- duces some hints and some ideas you may consider when choosing tests for convergence or divergence when evaluating a given series. It is usually a good idea to try using the Test for Divergence as a first step when evaluating a series for convergence or divergence. If we can show that:

lim n→∞ an 6 = 0

Then we can say that the series diverges without having to do any extra work.

Below are some general cases in which each test may help:

P-Series Test:

  • The series be written in the form:

np Geometric Series Test:

  • When the series can be written in the form:

anrn−^1 or

anrn

Direct Comparison Test:

  • When the given series, an looks like a known, or more simple, series, bn

Limit Comparison Test:

  • When you can see that the series looks like another convergent or divergent series, bn
  • But it is hard to say whether bn > an or bn < an

Root Test:

  • When the series can be written in the form:

(an)n

Alternating Series Test:

  • When the series can be written in the form:

(−1)n+1an or

(−1)nan

Ratio Test:

  • Whenever we are given something involving a factorial, e.g. n!
  • Whenever we are given something involving a constant raised to the nth^ power, e.g.

∑ (^) n+ 5 n Integral Test:

  • If the sequence is:
    • continuous
    • positive
    • decreasing (we can use the First Derivative Test here)

Calculus II Series - Things to Consider

Remember: These are just suggestions. There are other tests which may get us to the same answer. Convergent Divergent

∑^ ∞

n=

n^2 − 1 n^2 + n

∑^ ∞

n=

∑^ ∞

n=

3 n^3

∑^ ∞

n=

(−3)n+ 23 n

∑^ ∞

n=

(−3)n+ 23 n

∑^ ∞

n=

n^2 − 1 n^2 + n

∑^ ∞

n=

tan−^1 (n) n

n

∑^ ∞

n=

n^2 − 1 n^2 + n

∑^ ∞

n=

n^2 + n

∑^ ∞

n=

n^2 − 1 n^3 + 4

∑^ ∞

n=

( (^3) n 1 + 8n

)n ∑∞

n=

(n!)n n^4 n

∑^ ∞

n=

(−1)n^

n^2 − 1 n^2 + n

∑^ ∞

n=

(−1)n^

∑^ ∞

n=

2 nn! (n + 2)!

∑^ ∞

n=

∑^ ∞

n=

e (^1) n

n^2

∑^ ∞

n=

Test for Divergence n^ lim→∞ an^6 = 0

Test for Divergence n^ lim→∞ an^6 = 0

P-Series Test

P-Series Test

Geometric Series Test

Geometric Series Test

Direct Comparison Test

Direct Comparison Test

Limit Comparison Test

Limit Comparison Test

Root Test Root Test

Alternating Series Test

Alternating Series Test

Ratio Test Ratio Test

Integral Test Integral Test

Test for Divergence Fails n^ lim→∞ an^ = 0

Test for Divergence Fails n^ lim→∞ an^ = 0