Free Printable Test for Divergence

1.

$\sum_{n=1}^{\infty}\frac{n}{\sqrt{n^5}}$

a)

Converges

b)

Diverges

2.

$\sum_{n=1}^{\infty}\frac{n+10}{10n+1}$

a)

Converges

b)

Diverges

3.

$\sum_{n=1}^{\infty}\frac{6}{4^{3n}}$

a)

Converges

b)

Diverges

4.

$\sum_{n=1}^{\infty}ne^{-n}$

a)

Converges

b)

Diverges

5.

$\sum_{n=1}^{\infty}2\left(\frac{3}{2}\right)^n$

a)

Converges

b)

Diverges

6.

$\sum_{n=1}^{\infty}\frac{1}{n^{1.06}}$

a)

Converges

b)

Diverges

7.

$\sum_{n=1}^{\infty}\frac{1}{4^n}$

The series above is a

a)

geometric series

b)

p-series

8.

Let f be a positive, continuous, decreasing function such that

a_n=f\left(n\right)

. If

\sum_{n=1}^{\infty}a_n

converges to k, which of the following must be true?

a)

$\lim_{n\rightarrow\infty\ }a_n=k$

b)

$\int_1^nf\left(x\right)dx=k$

c)

$\int_1^{\infty}f\left(x\right)dx$ diverges

d)

$\int_1^{\infty}f\left(x\right)dx\$ converges

e)

$\int_1^{\infty}f\left(x\right)dx=k$

9.

Which of the following series diverge? (check all that apply)

a)

$\sum_{n=0}^{\infty}\left(\frac{\sin2}{\pi}\right)^n$

b)

$\sum_{n=1}^{\infty}\frac{1}{\sqrt{n}}$

c)

$\sum_{n=1}^{\infty}\frac{e^n}{e^n+1}$

10.

What is the sum of the converging geometric series below?

$\sum_{n=1}^{\infty}\frac{7^{n+1}}{10^n}$

(a)

11.

Which series diverges?

a)

$\sum_{n=1}^{\infty}\left(-1\right)^n\frac{1}{n}$

b)

$\sum_{n=1}^{\infty}\frac{1}{n^2}$

c)

$\sum_{n=1}^{\infty}\frac{1}{n^{\frac{1}{2}}}$

d)

$\sum_{n=1}^{\infty}7\left(\frac{4}{5}\right)^n$

12.

Given that $\int_1^{\infty}\left(\frac{1}{1+x^2}\right)dx=\frac{\pi}{4}$ , which of the following are true?

(check all that apply)

a)

$\sum_1^{\infty}\left(\frac{1}{1+x^2}\right)$ diverges

b)

$\sum_1^{\infty}\left(\frac{1}{1+x^2}\right)$ converges

c)

$\sum_1^{\infty}\left(\frac{1}{1+x^2}\right)=\frac{\pi}{4}$

13.

Which of the following series Diverge?
I. $\sum_{n=1}^{\infty}\frac{1}{n^2}$ II. $\sum_{n=1}^{\infty}\frac{1}{n}$ III. $\sum_{n=1}^{\infty}\frac{1}{\sqrt{n}}$

a)

I

b)

II

c)

II and III

d)

I and III

14.

Which of the following is true about the given series?

a)

it is a divergent p-series

b)

it is a convergent p-series

c)

it is a divergent geometric series

d)

it is a convergent geometric series

15.

Will the series converge or diverge?

\sum_{n=1}^{\infty}\frac{n+2}{\left(n+1\right)^3}

a)

converge

b)

diverge

16.

Will the series converge or diverge?

\sum_{n=1}^{\infty}\frac{\sqrt{3n-4}}{\sqrt{9n+2}}

a)

converge

b)

diverge

17.

Will the series converge or diverge?

\sum_{n=1}^{\infty}\frac{n!}{n\cdot3^n}

a)

converge

b)

diverge

18.

p-series:

A p-series will converge if __________ .

a)

$p<1$

b)

$p>1$

19.

Geometric Series:

A geometric series will converge if __________.

a)

$\left|r\right|>1$

b)

$\left|r\right|<1$

20.

nth term test:

If the limit as n approaches infinity of a series is __________ zero, then the series will diverge.

a)

Not equal to

b)

equal to

21.

Does the SERIES $\sum_{n=1}^{\infty}\frac{1}{4+e^{-x}}$ converge or diverge?

a)

Converge

b)

Diverge

22.

a)

Convergent

b)

Divergent

23.

a)

Convergent

b)

Divergent

24.

a)

Convergent

b)

Divergent

25.

a)

Divergent

b)

Convergent

26.

Does this sequence converge or diverge? $\frac{1}{3},1,3,9,...$

a)

Converges to 0

b)

Converges to 1

c)

Converges to 3

d)

Diverges

27.

Does this sequence converge or diverge? $\left\{\frac{10n-6}{2n-3}\right\}$

a)

Converges to 0

b)

Converges to 2

c)

Converges to 5

d)

Diverges

28.

Does this sequence converge or diverge? $\frac{1}{4},\frac{1}{9},\frac{1}{16},\frac{1}{25},...$

a)

Converges to 0

b)

Converges to 1

c)

Converges to 5

d)

Diverges

29.

An arithmetic sequence has a₁₅=160 and a common difference of -15. Find a₄₀.

a)

535

b)

370

c)

-170

d)

-215

30.

Geometric Series:

A geometric series will converge if __________.

a)

$\left|r\right|>1$

b)

$\left|r\right|<1$

31.

Find the 22nd term of the following sequence:
5, 8, 11, ...

a)

14

b)

68

c)

63

d)

71

32.

What is the explicit rule for the sequence:
11, 22, 44, 88...

a)

u_n = 2(11)^n-1

b)

u_n = 11(2)

c)

u_n = 11(2)^n-1

33.

a₄ = 3 and a₁₀ = -39

Find the common difference of the arithmetic sequence.

a)

-6

b)

-7

c)

-8

d)

-9

34.

a₃ = 225 and a₅ = 5625

Find the common ratio of the geometric sequence.

a)

3

b)

4

c)

5

d)

25

35.

Determine the Limit of the following sequence.

X_n=-nⁿ

a)

Diverges to inf

b)

Diverges to -inf

c)

Converges to 0

d)

The Limit Does Not Exist

36.

Determine the end behavior of the following sequence.

X_n=1/(2n)

a)

Converges to 0

b)

Convergese to 1/2

c)

Diverges to inf

d)

Diverges to -inf

37.

Decide which of the following Sequences Converge to 0

a)

{1, 1, 1, 1, ...}

b)

$\frac{1}{n+1}$

c)

$-1^n$

d)

$\frac{n}{2}$

38.

An Arithmetic Sequence can Converge to a specific number.

a)

True, if the sequence is finite.

b)

False, it can only converge to -inf, inf, or 0.

39.

a)

Divergent

b)

Convergent

40.

*hint - you will need to use an adaption of the definition of e when you take your limit.

a)

Convergent

b)

Divergent

41.

$\sum_{n=1}^{\infty}\frac{n}{\sqrt{n^5}}$

a)

Converges

b)

Diverges

42.

$\sum_{n=1}^{\infty}\frac{n+10}{10n+1}$

a)

Converges

b)

Diverges

43.

$\sum_{n=1}^{\infty}2\left(\frac{3}{2}\right)^n$

a)

Converges

b)

Diverges

44.

Let f be a positive, continuous, decreasing function such that

a_n=f\left(n\right)

. If

\sum_{n=1}^{\infty}a_n

converges to k, which of the following must be true?

a)

$\lim_{n\rightarrow\infty\ }a_n=k$

b)

$\int_1^nf\left(x\right)dx=k$

c)

$\int_1^{\infty}f\left(x\right)dx$ diverges

d)

$\int_1^{\infty}f\left(x\right)dx\$ converges

e)

$\int_1^{\infty}f\left(x\right)dx=k$

45.

Which of the following series diverge? (check all that apply)

a)

$\sum_{n=0}^{\infty}\left(\frac{\sin2}{\pi}\right)^n$

b)

$\sum_{n=1}^{\infty}\frac{1}{\sqrt{n}}$

c)

$\sum_{n=1}^{\infty}\frac{e^n}{e^n+1}$

46.

What is the sum of the converging geometric series below?

$\sum_{n=1}^{\infty}\frac{7^{n+1}}{10^n}$

(a)

47.

Which statement is true about the SERIES $\sum_{n=1}^{\infty}\frac{6}{n\sqrt[]{n}}$ ?

a)

It is a convergent geometric series.

b)

It is a convergent P-series.

c)

It is a divergent P-series.

d)

It is a divergent geometric series.

48.

nth term test:

If the limit as n approaches infinity of a series is __________ zero, then the series will diverge.

a)

Not equal to

b)

equal to

49.

Geometric Series:

A geometric series will converge if __________.

a)

$\left|r\right|>1$

b)

$\left|r\right|<1$

50.

p-series:

A p-series will converge if __________ .

a)

$p<1$

b)

$p>1$

51.

Will the series converge or diverge?

\sum_{n=1}^{\infty}\frac{n!}{n\cdot3^n}

a)

converge

b)

diverge

52.

Will the series converge or diverge?

\sum_{n=1}^{\infty}\frac{\sqrt{3n-4}}{\sqrt{9n+2}}

a)

converge

b)

diverge

53.

Will the series converge or diverge?

\sum_{n=1}^{\infty}\frac{3^n}{8^{\left(n+1\right)}}

a)

converge

b)

diverge

54.

Which of the following tests would be the best choice to prove the series is divergent?

a)

nth term test

b)

integral test

c)

direct comparison test

d)

limit comparison test

55.

Which statement is true about the series?

a)

The sum is -4/3

b)

The sum is 2/3

c)

The sum is 9/2

d)

The series diverges

56.

Which of the following is true about the given series?

a)

it is a divergent p-series

b)

it is a convergent p-series

c)

it is a divergent geometric series

d)

it is a convergent geometric series

57.

Select the correct comparison series you would use for the given series.

a)

$\Sigma\ \frac{1}{n}$

b)

$\Sigma\ \frac{1}{n^2}$

c)

$\Sigma\ \left(\frac{1}{2}\right)^n$

d)

$\Sigma\ \left(2\right)^n$

58.

What would be the test used, and what is the result of the test?

a)

alternating series, absolutely converge

b)

nth term test, diverge

c)

alternating series, diverge

d)

integral test, converge

59.

Which test could be used to show that the series $\sum_{n=2}^{\infty}\frac{1}{n\ln n}$ diverges?

a)

Geometric Series Test

b)

p-Series Test

c)

Integral Test

d)

nth-Term Test for Divergence

60.

$\sum_{n=1}^{\infty}\frac{n}{\sqrt{n^5}}$

a)

Converges

b)

Diverges

61.

$\sum_{n=1}^{\infty}\frac{n+10}{10n+1}$

a)

Converges

b)

Diverges

62.

$\sum_{n=1}^{\infty}\frac{6}{4^{3n}}$

a)

Converges

b)

Diverges

63.

$\sum_{n=1}^{\infty}ne^{-n}$

a)

Converges

b)

Diverges

64.

$\sum_{n=1}^{\infty}2\left(\frac{3}{2}\right)^n$

a)

Converges

b)

Diverges

65.

$\sum_{n=1}^{\infty}\frac{1}{n^{1.06}}$

a)

Converges

b)

Diverges

66.

$\sum_{n=1}^{\infty}\frac{1}{4^n}$

The series above is a

a)

geometric series

b)

p-series

67.

Let f be a positive, continuous, decreasing function such that

a_n=f\left(n\right)

. If

\sum_{n=1}^{\infty}a_n

converges to k, which of the following must be true?

a)

$\lim_{n\rightarrow\infty\ }a_n=k$

b)

$\int_1^nf\left(x\right)dx=k$

c)

$\int_1^{\infty}f\left(x\right)dx$ diverges

d)

$\int_1^{\infty}f\left(x\right)dx\$ converges

e)

$\int_1^{\infty}f\left(x\right)dx=k$

68.

Which of the following series diverge? (check all that apply)

a)

$\sum_{n=0}^{\infty}\left(\frac{\sin2}{\pi}\right)^n$

b)

$\sum_{n=1}^{\infty}\frac{1}{\sqrt{n}}$

c)

$\sum_{n=1}^{\infty}\frac{e^n}{e^n+1}$

69.

What is the sum of the converging geometric series below?

$\sum_{n=1}^{\infty}\frac{7^{n+1}}{10^n}$

(a)

70.

Which of the following series should you use the ratio test to determine convergence or divergence?

a)

$\sum_{n=1}^{\infty}\frac{n^2+n}{3n^2+5}$

b)

$\sum_{n=1}^{\infty}\frac{n^n}{n!}$

c)

$\sum_{n=1}^{\infty}\frac{\ln\ n}{n}$

d)

$\sum_{n=1}^{\infty}\frac{1}{2^n}$

71.

Which series diverges by the nth term test?

a)

$\sum_{n=1}^{\infty}\frac{n}{n!}$

b)

$\sum_{n=1}^{\infty}\frac{3^n}{5^n}$

c)

$\sum_{n=1}^{\infty}\frac{5n^2}{3n^3+n^2}$

d)

$\sum_{n=1}^{\infty}\frac{1}{\tan^{-1}n}$

72.

Which series can the integral test be used to determine convergence or divergence?

a)

$\sum_{n=1}^{\infty}\frac{n}{e^{n^2}+n}$

b)

$\sum_{n=1}^{\infty}\frac{\ln n}{n}$

c)

$\sum_{n=1}^{\infty}\frac{\sin^{-1}n}{\sqrt{1-n^2}}$

d)

all of the above

Test for Divergence

Total questions: 72

Worksheet time: 3570secs

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