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

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Question 10

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Question 11

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Question 12

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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

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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

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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

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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