Series and Tests for Series

Question 1

Telescoping Series: Use definition (2) to see if a telescoping series converges or diverges.

Answer 1

Expand S_n , collapse the terms, and take the limit as n→∞

If the limit exists, then the series converges to this limit.

If the limit does not exist, the series diverges.

Question 2

Answer 2

Question 3

The converse of Theorem 6

Answer 3

Question 4

Answer 4

Question 5

Answer 5

Question 6

Answer 6

Question 7

Does a Harmonic Series Converge or Diverge?

Answer 7

Question 8

Answer 8

Question 9

Answer 9

Question 10

Answer 10

Question 11

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Question 12

Answer 12

Question 13

1 / increasing function is

Answer 13

a decreasing function

Question 14

The second method to prove a function is decreasing?

Answer 14

Show f’ (x) <0 eventually

Question 15

Answer 15

Question 16

Answer 16

Question 17

Answer 17

Question 18

A Useful Comparison for

n >3

is?

Answer 18

ln n >1

Question 19

A Useful Comparison for a>0

is?

Answer 19

ln n < n^a

Question 20

A Useful Comparison for

e ^1/<em>n</em>

Is it is between?

Answer 20

1< e ^{1/<em>n </em>}< e

Question 21

A Useful Comparisons for tan^-1

is it falls between

Answer 21

0-1 n < ∏/2 <2

Question 22

A Useful Comparison for sin² x is it falls between?

Answer 22

0< sin² x < 1

Question 23

A Useful Comparisons for cos²x

is it falls between?

Answer 23

0 < cos²x < 1

Question 24

Answer 24

Question 25

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Question 26

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Question 27

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Question 28

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Question 29

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Question 30

What is the 3 Step procedure when asked if a series is **absolutely** convergent, **conditionally** convergent, or **diverges**?

Answer 30

**Step 1**: **Divergence test** in your head on the positive part. If the positive terms do not → 0, the series diverges by the DIVERGENCE TEST Stop. If the terms → 0,, move to Step 2. * *Step 2**: **Check absolute convergence**. If the series converges, it converges absolutely (and hence converges) and you are done. If no, move to Step 3. * *Step 3**: Check for **Conditional Convergence** (with the alternating series test). If the series converges but not absolutely, then the series is conditionally convergent. *We do not have any other resource in this course for **non-positive** series, so you must be given an alternating series if you have arrived at* Step 3.

Question 31

Answer 31

Question 32

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Question 33

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Question 34

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Question 35

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Question 36

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Question 37

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Question 38

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Question 39

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Question 40

Answer 40