# Calculus 2 Infinite Series & Convergence Tests

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When is a geometric series convergent?

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### 30 Terms

1

When is a geometric series convergent?

|r|<1

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2

When is a geometric series divergent?

|r|>=1

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3

What is the sum of a geometric series?

a/(1-r) , where a is the first term

* can only find the sum if the geometric series is convergent

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4

When is a p series convergent?

p > 1

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5

When is a p series divergent?

p <= 1

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6

When is a series divergent by the test for divergence?

lim n→ (an) is nonzero or doesn’t exist

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7

When is the test for divergence inconclusive?

lim n→ (an) = 0

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8

What are the conditions for using the integral test?

after rewriting an as f(x), f must be

1. continuous

2. positive

3. decreasing

on [n, ∞)

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9

When does a series converge by the integral test?

n∞ f(x) dx converges

(lim t→n t f(x) dx is a constant)

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10

When does a series diverge by the integral test?

n∞ f(x) dx diverges

(lim t→n t f(x) dx is undefined or goes to infinity)

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11

When does a series Σan converge by the direct comparison test?

Σbn converges and an <= bn

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12

When does a series Σan diverge by the direct comparison test?

Σbn diverges and an >= bn

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13

what can we conclude from the limit comparison test?

If lim n→∞ an / bn = finite number > 0, Σbn and Σan have the same convergence behavior (either both converge or both diverge)

* choose Σbn to be a simpler series whose convergence behavior we can figure out, usually a p series or geometric series

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14

when is the limit comparison test inconclusive?

lim n→∞ an / bn = 0 or ∞

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15

When does an alternating series Σ(-1)^n * bn converge by the alternating series test?

1. bn+1 <= bn ({bn} is decreasing)

2. lim n→∞ bn = 0

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16

When is the alternating series test inconclusive?

At least one of the following conditions is violated:

1. bn+1 <= bn ({bn} is decreasing)

2. lim n→∞ bn = 0

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17

When is a series Σan absolutely convergent?

Σ|an| converges (and therefore Σan converges too)

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18

When is a series Σan conditionally convergent?

Σ|an| diverges and Σan converges

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19

When is a series Σan absolutely convergent (and therefore convergent) by the ratio test?

lim n→∞ | an+1 / an | is a finite number <1

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20

When is a series Σan divergent by the ratio test?

lim n→∞ | an+1 / an | > 1 or is ∞

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21

When is the ratio test inconclusive?

lim n→∞ | an+1 / an | = 1

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22

When is a series Σan absolutely convergent (and therefore convergent) by the root test?

lim n→n√|an| is a finite number <1

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23

When is a series Σan divergent by the root test?

lim n→n√|an| > 1 or is ∞

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24

When is the root test inconclusive?

lim n→n√|an| = 1

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25

When is a sequence {an} convergent?

lim n→∞ an = a finite number

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26

When is a sequence {an} divergent?

lim n→∞ an doesn’t exist (undefined or goes to ∞)

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27

what are the conditions for using the direct comparison test?

all terms of an and bn must be > 0

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28

what are the conditions for using the limit comparison test?

all terms of an and bn must be > 0

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29

what condition must be met to use the ratio test?

all terms of the series are nonzero

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30

what conditions must be met to use the alternating series test?

1. has the form Σ(-1)^n * bn or Σ(-1)^(n+a number) * bn

2. bn > 0

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