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Hint

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→∞ (a_{n}) is nonzero or doesn’t exist

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7

When is the test for divergence inconclusive?

lim n→∞ (a_{n}) = 0

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8

What are the conditions for using the integral test?

after rewriting a_{n }as f(x), f must be

continuous

positive

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 Σa_{n} converge by the direct comparison test?

Σb_{n }converges and a_{n }<= b_{n}

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12

When does a series Σa_{n} diverge by the direct comparison test?

Σb_{n }diverges and a_{n }>= b_{n}

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13

what can we conclude from the limit comparison test?

If lim n→∞ a_{n }/ b_{n }= finite number > 0, Σb_{n }and Σa_{n }have the same convergence behavior (either both converge or both diverge)

* choose Σb_{n }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→∞ a_{n }/ b_{n }= 0 or ∞

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15

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

b

_{n+1 }<= b_{n }({b_{n}}_{ }is decreasing)lim n→∞ b

_{n }= 0

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16

When is the alternating series test inconclusive?

At least one of the following conditions is violated:

b

_{n+1 }<= b_{n }({b_{n}}_{ }is decreasing)lim n→∞ b

_{n }= 0

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17

When is a series Σa_{n }absolutely convergent?

Σ|a_{n}| converges (and therefore Σa_{n }converges too)

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18

When is a series Σa_{n }conditionally convergent?

Σ|a_{n}| diverges and Σa_{n }converges

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19

When is a series Σa_{n }absolutely convergent (and therefore convergent) by the ratio test?

lim n→∞ | a_{n+1} / a_{n} | is a finite number <1

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20

When is a series Σa_{n }divergent by the ratio test?

lim n→∞ | a_{n+1} / a_{n} | > 1 or is ∞

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21

When is the ratio test inconclusive?

lim n→∞ | a_{n+1} / a_{n} | = 1

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22

When is a series Σa_{n }absolutely convergent (and therefore convergent) by the root test?

lim n→∞ ^{n}√|a_{n}| is a finite number <1

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23

When is a series Σa_{n }divergent by the root test?

lim n→∞ ^{n}√|a_{n}| > 1 or is ∞

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24

When is the root test inconclusive?

lim n→∞ ^{n}√|a_{n}| = 1

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25

When is a sequence {a_{n}} convergent?

lim n→∞ a_{n} = a finite number

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26

When is a sequence {a_{n}} divergent?

lim n→∞ a_{n} 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 a_{n }and b_{n }must be > 0

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28

what are the conditions for using the limit comparison test?

all terms of a_{n }and b_{n }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?

has the form Σ(-1)^n * b

_{n}or Σ(-1)^(n+a number) * b_{n}b

_{n }> 0

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