Deck 11: Infinite Series

Question 1

Let be defined recursively by .
Which of the following statements holds?

A) If , then .
B) If , then .
C) If , then .
D) If , then .
E) If , then

Answer 1

If , then .

Question 2

Find the sum of the following series. (Use the sum , as necessary.)

A)
B)

Answer 2

A) B)

Question 3

Use the Squeeze Theorem to evaluate the limits.

A)
B)
C) (Hint: use .)

Answer 3

A) 7
B) 1
C) 1

Question 4

Evaluate .

Question 5

Use the Squeeze Theorem to evaluate the limits.

A)
B)
C) (Hint: use .)

Question 6

Find the partial sum of the series and determine the sum of the series.

Question 7

Evaluate the limits.

A)
B) (Hint: simplify the sequence.)
C)

Question 8

Let be defined recursively by .
Which of the following statements holds?

A) If , then .
B) If , then .
C) If , then .
D) If then .
E) None of the above is correct.

Question 9

Find the partial sums of the series , and determine its sum.

Question 10

Find the sum of the following series.

A)
B)
C)

Question 11

Let be a positive sequence such that .
Which of the following statement holds?

A) The sequence is decreasing but does not converge.
B) The sequence is decreasing and converging to .
C) The sequence is increasing and diverging.
D) The sequence is decreasing and converging to .
E) None of the above is correct.

Question 12

Find the partial sums of the series , and determine if the series converges.

Question 13

Let be defined recursively by , .
Which of the following statements holds?

A) is increasing and converging to 1.
B) is decreasing and converging to 1.
C) is decreasing and converging to 5.
D) is increasing and converging to 5.
E) None of the above is correct.

Question 14

Let be defined recursively by , . Which of the following statements holds?

A) is decreasing and converging to
B) is increasing and unbounded.
C) is increasing and converging to 3.
D) is not monotonic by converging to 3.
E) is decreasing and unbounded.

Question 15

Let be a sequence such that .
Use the definition of the limit and the Squeeze Theorem to find .

Question 16

converges, and if so find its limit. Use the following fact: if , then .
Ans: is increasing with upper bound, hence converging.
Let be defined recursively by , Which of the following statements holds?

A) is decreasing and converging to
B) is decreasing to
C) is increasing and converging to
D) is decreasing and converging to
E) is increasing to

Question 17

Let be defined recursively by , .
Which of the following statement holds?

A) is increasing and converging to 1.
B) is decreasing and converging to 1.
C) is decreasing and converging to 5.
D) is increasing and converging to 5.
E) None of the above is correct.

Question 18

Evaluate the limits.

A)
B)
C)

Question 19

Which of the following statements is (are) correct for

A) The series converges since the partial sums are zero.
B) The series converges since the partial sums are 1 and 0.
C) The series diverges since the partial sums are 1 and 0.
D) The series diverges since the general term does not tend to zero.
E) Answers C and D are correct.

Question 20

Evaluate the following limits.

A)
B)
C)

Question 21

Let . Determine if the series converges.

Question 22

Find the partial sums of the series , and determine its sum.

Question 23

Use the Comparison Tests to determine if the following series converge.

A)
B)
C)

Question 24

Evaluate

Question 25

Use the Direct Comparison Test or the Limit Comparison Test to determine if the series is convergent.

A)
B)
C)

Question 26

Determine if the following series converge.

A)
B)
C)

Question 27

Verify that the conditions for the Integral Test are satisfied, and use it to determine if the series is convergent.

A)
B)
C)

Question 28

Find the partial sums of the series , and determine the sum of the series if it is convergent.

Question 29

Determine whether the series converges.

Question 30

Find the partial sums of , and determine if the series converges.

Question 31

Evaluate .

Question 32

Evaluate .

Question 33

Use the Integral Test to determine if the following series converge.

A)
B)

Question 34

Determine if the following series converge.

A)
B)
C)

Question 35

Find the partial sums of the series , and determine its sum.

Question 36

Which of the following series diverges?

A)
B)
C)
D)
E) B and D

Question 37

Determine if the following series converge.

A)
B)
C)

Question 38

Use the Integral Test and Integration by Parts to determine if the following series converges.

Question 39

Determine if the following series converge.

A)
B)
C)

Question 40

Determine if the following series converge.

A)
B)
C)

Question 41

Determine if the following series is conditionally convergent, absolutely convergent, or divergent.

A)
B)

Question 42

Determine if the following series is conditionally convergent, absolutely convergent, or divergent.

A)
B)
C)

Question 43

Determine whether the series is absolutely convergent, conditionally convergent, or divergent.

A)
B)
C)

Question 44

Determine whether the series converges or diverges.

Question 45

Determine whether the series is conditionally convergent, absolutely convergent, or divergent.

A)
B)
C)

Question 46

Which of the following statements is incorrect?

A) If converges, then also converges.
B) If converges, then also converges.
C) If is bounded and is absolutely convergent then is absolutely convergent.
D) If , then is absolutely convergent.
E) None of the above is correct.

Question 47

a) Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
b) If the series converges to , find such that .

Question 48

Consider the series .

A) Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
B) If the series converges to , find such that .
(Hint: use the inequality .)

Question 49

Which one(s) of the following series is(are) conditionally convergent?

A)
B)
C)
D)
E) C and D

Question 50

Which of the following series is conditionally convergent?

A)
B)
C)
D)
E) B and D

Question 51

Use the Comparison Tests to determine if the following series converge.

A)
B)
C)

Question 52

Determine whether the series converges or diverges.

Question 53

Determine whether the series converges or diverges.

Question 54

Determine whether the series converges or diverges. If it converges, does it converge conditionally or absolutely?

Question 55

Determine whether the series is absolutely convergent, conditionally convergent, or divergent.

A)
B)
C)

Question 56

Which of the following is true?

A) If converges conditionally, then converges absolutely.
B) If converges absolutely, then converges absolutely.
C) both (A) and (B)
D) neither (A) nor (B)

Question 57

Use the Integral Test to determine whether the series converges.

Question 58

Apply the Ratio Test to determine if the series is convergent.

A)
B)
C)

Question 59

Use the Root Test to determine if the series is convergent.

A)
B)
C) (Hint: use .)
D)

Question 60

Determine whether the series converges or diverges. If it converges, does it converge conditionally or absolutely?

Question 61

The radius of convergence of is:

A) .
B) .
C) .
D) .
E) .

Question 62

Use the Root Test to determine if the following series converge.

A)
B)
C)

Question 63

Does the series converge or diverge?

Question 64

Use the Ratio Test to determine if the following series converge.

A)
B)
C)

Question 65

Use the formula for geometric series to expand the function in a power series with center . Determine the values of for which the expansion is valid.

Question 66

Find the values of for which the series converges.

Question 67

The values of for which the series converges are:

A) .
B) .
C) .
D) .
E) .

Question 68

Use the Ratio Test to determine if the following series converge.

A)
B)
C)

Question 69

Does the series converge or diverge?

Question 70

The values of at which the series converges are:

A) .
B) .
C) .
D) .
E) .

Question 71

Does the series ?converge or diverge

Question 72

Use the Root Test to determine if the following series converge.

A) ( )
B)
C)

Question 73

Find the values of for which the series converges.

Question 74

Determine whether the series is conditionally convergent, absolutely convergent, or divergent.

A)
B)

Question 75

Let .

A) Find the values of for which the series is convergent.
B) Find the series for .

Question 76

Does the series converge or diverge?

Question 77

Does the series converge or diverge?

Question 78

Find the exact intervals on which the series converge.

A)
B)
C)

Question 79

Use the Root Test to determine if the following series converge.

A)
B)
C)

Question 80

If , then which of the following holds?

A) The series converges.
B) The series converges.
C) If converges, then it converges absolutely.
D) The sequence of partial sums of is bounded.
E) B and D both hold.