Deck 10: Sequences, Series, and Power Series

Question 1

Which of the following descriptors apply to the sequence ?
(a) increasing (or ultimately increasing)
(b) decreasing (or ultimately decreasing)
(c) positive (or ultimately positive)
(d) negative (or ultimately negative)
(e) bounded below only
(f) bounded above only
(g) bounded
(h) unbounded above and below
(i) alternating
(j) divergent (but not to $\infty$ or - $\infty$ )
(k) divergent to $\infty$
(l) divergent to - $\infty$
(m) convergent

A) (a), (c), (e), (m)
B) (a), (c), (f), (j)
C) (a), (c), (g), (m)
D) (b), (c), (g), (l)
E) (a), (c), (e), (k)

Answer 1

(a), (c), (g), (m)

Question 2

Find the limit of the sequence .

A) 0
B) 1
C) 2
D) 3
E)

Answer 2

2

Question 3

Which of the following descriptors apply to the sequence ?

A) (g), (i), (m)
B) (b), (f), (m)
C) (g), (i), (j)
D) (b), (c), (i), (l)
E) (g), (i), (k)

Answer 3

(g), (i), (m)

Question 4

Find the limit of the sequence .

A) 0
B) 1
C) 2
D) 3
E) -1

Question 5

Which of the following descriptors apply to the sequence ?
(a) increasing (or ultimately increasing)
(b) decreasing (or ultimately decreasing)
(c) positive (or ultimately positive)
(d) negative (or ultimately negative)
(e) bounded below only
(f) bounded above only
(g) bounded
(h) unbounded above and below
(i) alternating
(j) divergent (but not to $\infty$ or - $\infty$ )
(k) divergent to $\infty$
(l) divergent to - $\infty$
(m) convergent

A) (g), (i), (m)
B) (a), (f), (m)
C) (g), (i), (j)
D) (b), (c), (i), (l)
E) (g), (i), (k)

Question 6

Which of the following descriptors apply to the sequence ?
(a) increasing (or ultimately increasing)
(b) decreasing (or ultimately decreasing)
(c) positive (or ultimately positive)
(d) negative (or ultimately negative)
(e) bounded below only
(f) bounded above only
(g) bounded
(h) unbounded above and below
(i) alternating
(j) divergent (but not to $\infty$ or - $\infty$ )
(k) divergent to $\infty$
(l) divergent to - $\infty$
(m) convergent

A) (a), (d), (f), (l)
B) (a), (f), (m)
C) (a), (c), (e), (k)
D) (b), (f), (j)
E) (b), (d), (f), (l)

Question 7

Let = , n = 1, 2, 3,...
Which of the following statements about the sequence is true?

A) The sequence is decreasing with a lower bound equal to .
B) The sequence is increasing with an upper bound equal to .
C) The sequence is increasing with a lower bound equal to 0.
D) The sequence is not monotonic and also unbounded.
E) The sequence is decreasing with an upper bound equal to .

Question 8

Find the limit of the sequence .

A) 0
B) 1
C) 2
D) 3
E) 6

Question 9

The sequence is defined recursively by = , = , n = 1, 2, 3,...Assuming the sequence converges to the real number L, find L.

A) 1
B) -4 or 1
C) -1 or 4
D) -1
E) 4

Question 10

Find the limit of the sequence { }, where = - .

A) -54
B) 54
C) - $\infty$
D) 0
E) $\infty$

Question 11

Show that the sequence converges, and find its limit.

Question 12

Find the limit of the sequence .

A) 0
B) 1
C) 2
D) 3
E) This sequence is divergent.

Question 13

Find, if it exists, the limit of the sequence { }, where a_n = .

A) 0
B) 1
C) 2
D) 3
E) The limit does not exist.

Question 14

The sequence

A) diverges to - $\infty$ .
B) converges to 0.
C) converges to 3.
D) diverges to + $\infty$ .
E) converges to - 4.

Question 15

Find the limit of the sequence .

A) 9
B) 3
C) -3
D) 0
E) -9

Question 16

If {| |} converges, then { } converges.

Question 17

If { } converges, then {| |} converges.

Question 18

If { } converges, then must diverge.

Question 19

If { } converges and { } converges, then must converge.

Question 20

= is the partial sum of an infinite series . Find . Determine if the series is convergent or divergent, and if convergent find its sum.

A) = , = - for n $\ge$ 2; series converges, sum is 0
B) = for n $\ge$ 1; series converges, sum is 1
C) = , = - for n $\ge$ 2; series converges, sum is
D) = for n $\ge$ 1; series converges, sum is
E) = for n $\ge$ 1; series converges, sum is

Question 21

= is the partial sum of an infinite series . Find . Determine if the series is convergent or divergent, and if convergent find its sum.

A) = , series diverges to $\infty$

B) = , series diverges to $\infty$

C) = , series converges to 1
D) = , series converges to 2
E) = , series converges to 1

Question 22

= is the partial sum of an infinite series . Find . Determine if the series is convergent or divergent, and if convergent find its sum.

A) = , series converges to
B) = , series converges to 1
C) = , series converges to 2
D) = , series diverges to $\infty$

E) = , series diverges to $\infty$

Question 23

Find the sum of the series 4 - 1 + - +...

A)
B)
C)
D)
E)

Question 24

On what interval of values of x does the series converge? What is its sum for x in that interval?

A) (-1, 1), sum =
B) (-1, 1), sum =
C) (-1, 1], sum =
D) [-1, 1), sum =
E) [-1, 1], sum =

Question 25

Find the sum of the series + + +... + +... = .

A)
B) 1
C) 2
D)
E) $\infty$ (series diverges)

Question 26

The geometric series

A) converges and its sum is .
B) converges and its sum is .
C) diverges and hence it has no sum.
D) converges and its sum is 4.
E) converges and its sum is .

Question 27

Find the sum of the series + + +... + +... = .

A)
B)
C) 1
D) $\infty$ (series diverges)
E) none of the above

Question 28

Find the sum of the series

A)
B)
C)
D)
E) $\infty$ (series diverges)

Question 29

Evaluate .

A)
B)
C)
D)
E) series diverges

Question 30

Evaluate .

A) e
B) 1
C)
D)
E) $\infty$ (series diverges)

Question 31

Show that the infinite series converges, and find its sum.Hint : Verify that = - .

Question 32

Evaluate .

A) -1
B) 0
C) 1
D)
E) no sum, series diverges

Question 33

Find the sum .

A)
B)
C) 1
D) $\infty$ (series diverges)
E) none of the above

Question 34

Find the sum of the series .

A)
B)
C)
D)
E) $\infty$ (series diverges)

Question 35

Find the sum of the series .

A) 2
B) 3
C) 1
D) 4
E) $\infty$ (series diverges)

Question 36

If diverges to infinity and $\neq$ 0 for all n, then converges.

Question 37

If and both diverge, then also diverges.

Question 38

If converges, then = 0.

Question 39

Question 40

The series converges.

Question 41

The series converges.

Question 42

The series converges.

Question 43

The series converges.

Question 44

For exactly what values of the constant k does the series converge?

A) p > 2
B) p $\ge$ 2
C) p < 1
D) p $\le$ 1
E) p > 1

Question 45

Find all values of the nonzero constant real number p such that the series is convergent.

A) p > 1
B) 0 < P $\le$ 1
C) 0 < P $\le$
D) p < 0
E) p > 0

Question 46

For exactly what values of the constant p does the series converge?

A) p $\ge$ 1
B) p > 1
C) p $\ge$ 2
D) p > 2
E) p > 0

Question 47

For exactly what values of the constant p does the series converge?

A) -3 < p < 3
B) -3 $\le$ p $\le$ 3
C) 0 $\le$ p $\le$ 3
D) 0 < p < 3
E) -3 < p < 0

Question 48

The P-series , x > 0 converges for all x such that:

A) x > 1
B) x $\ge$ 1
C) 0 < x < 1
D) 0 < x $\le$ 1
E) x (- $\infty$ , $\infty$ )

Question 49

The series converges.

Question 50

The series converges.

Question 51

The series converges.

Question 52

Use the integral test bounds to estimate the sum of the series using the first three terms.

A) 104
B) 140
C) 119
D) 101
E) 98

Question 53

What is the largest positive constant K such that if 0 < p < K, then must converge?

A) 1
B) 2
C) 4
D) 8
E) 9

Question 54

Give an upper bound for the error when is approximated by the partial sum .

A) 0.0004
B) 0.0001
C) 0.00004
D) 0.00001
E) 0.004

Question 55

Give an upper bound for the error when e = is approximated by the partial sum . How large should n be taken to ensure that the error is less than 0.001?

A) , n = 7
B) , n = 8
C) , n = 7
D) , n = 8
E) , n = 6

Question 56

The series converges.

Question 57

Question 58

The series converges.

Question 59

The series converges.

Question 60

The series converges.

Question 61

The series converges.

Question 62

The series converges.

Question 63

The series is an alternating series.

Question 64

For what values of x does the series
(a) converge absolutely?
(b) converge conditionally?

A) (a) x = 0 only, $~~~~~~~~$ (b) nowhere
B) (a) -1 < x < 1, $~~~~~~~~$ (b) nowhere
C) (a) -1 < x < 1, $~~~~~~~~$ (b) x = -1 and x = 1
D) (a) all real x, $~~~~~~~~$ (b) nowhere
E) (a) -1 < x < 1, $~~~~~~~~$ (b) x = 1

Question 65

For what values of x does the series
(a) converge absolutely?
(b) converge conditionally?

A) (a) on the interval (1, 5), $~~~~~~~~$ (b) at x = 1
B) (a) on the interval (1, 5), $~~~~~~~~$ (b) at x = -1 and x = 5
C) (a) on the interval [1, 5], $~~~~~~~~$ (b) nowhere
D) (a) on the interval [1, 5), $~~~~~~~~$ (b) at x = 5
E) (a) all real x, $~~~~~~~~$ (b) nowhere

Question 66

For what values of x does the series
(a) converge absolutely?
(b) converge conditionally?

A) (a) on the interval (-2, -1), $~~~~~~~~$ (b) at x = -2 and x = -1
B) (a) on the interval [-2, -1] $~~~~~~~~$ (b) nowhere
C) (a) on the interval (-2, -1), $~~~~~~~~$ (b) for x < -2
D) (a) on the interval (-3, 0), $~~~~~~~~$ (b) at x = -3 and x = 0
E) (a) on the interval [-2, -1), $~~~~~~~~$ (b) at x = -1

Question 67

Which of the three series (i) (ii) (iii)
is conditionally convergent?

A) series (iii)
B) series (i)
C) series (i) and (iii)
D) series (ii)
E) All three

Question 68

Which of the three series (i) (ii) (iii) is absolutely convergent?

A) series (i) and (ii)
B) series (i)
C) series (ii) and (iii)
D) series (i) and (iii)
E) none of the series

Question 69

For what values of x does the series
converge?

A) on the interval [1, $\infty$ )
B) on the intervals (- $\infty$ , -1] and [1, $\infty$ )
C) for all x $\neq$ 0
D) for x = 2 only
E) on the interval (- $\infty$ , -1]

Question 70

For what values of x does the series converge?

A) on the intervals (- $\infty$ , -1] and [1, $\infty$ )
B) on the intervals (- $\infty$ , -1) and (1, $\infty$ )
C) on the interval [-1, 1]
D) on the interval (-1, 1)
E) on the interval [1, $\infty$ )

Question 71

For what values of x does the series ln x + + + +... converge?

A) < x < e
B) 0 < x < e
C) 1 < x < e
D) 0 < x < $\infty$
E) 0 < x < 1

Question 72

If is conditionally convergent, then must be divergent.

Question 73

If is divergent, then must be conditionally convergent.

Question 74

By possibly rearranging its terms, the series can be forced to have the sum 17.

Question 75

Find the radius, centre, and interval of convergence of the series .

A) centre 0, radius 1, interval [-1, 1)
B) centre 0, radius 1, interval [-1, 1]
C) centre 0, radius 1, interval (-1, 1)
D) centre 0, radius $\infty$ , interval (- $\infty$ , $\infty$ )
E) centre 0, radius 1, interval (-1, 1]

Question 76

Find the radius, centre, and interval of convergence of the series .

A) centre 1, radius , interval
B) centre 1, radius , interval
C) centre 1, radius , interval
D) centre 1, radius , interval
E) centre 1, radius , interval ,

Question 77

The interval of convergence of the power series is given by [-7, -3).Find the centre c and the radius of convergence R of the power series.

A) c = -10, R = 4
B) c = -5, R = 2
C) c = -4, R = 10
D) c = -2, R = 5
E) c = 2, R = -5

Question 78

Find the centre, radius, and interval of convergence of the series

A) centre 0, radius 1, interval [-1, 1]
B) centre 0, radius 1, interval (-1, 1]
C) centre 0, radius 1, interval (-1, 1)
D) centre 0, radius 2, interval (-2, 2)
E) centre 0, radius 1, interval [-1, 1)

Question 79

Find the centre, radius, and interval of convergence of the series

A) centre 4, radius 3, interval (1, 7]
B) centre 4, radius 3, interval [1, 7)
C) centre 4, radius 1, interval [3, 5]
D) centre 4, radius 1, interval (3, 5)
E) centre 4, radius 1, interval (3, 5]

Question 80

Find the centre, radius, and interval of convergence of the series

A) centre -3, radius 1, interval [-4, -2]
B) centre -3, radius 1, interval (-4, -2)
C) centre 0, radius $\infty$ , interval (- $\infty$ , $\infty$ )
D) centre -3, radius $\infty$ , interval (- $\infty$ , $\infty$ )
E) centre -3, radius , interval