Free Printable Unit 10A Review

1.

Does the sequence

$a_n=\frac{3n}{2n-1}$ converge or diverge?

a)

Converge

b)

Diverge

2.

Does the series $\sum_{n=1}^{\infty}\frac{3n}{2n-1}$ converge or diverge?

a)

Converge

b)

Diverge

3.

What conclusion can you make using the nth term test to describe

\sum_{n=1}^{\infty}\frac{6n}{4n^2+1}

a)

This series diverges by the nth term test

b)

This series converges by the nth term test

c)

No conclusion about this series can be made using the nth term test.

4.

Determine the convergence/divergence of the series

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

and find the sum (if possible).

a)

This series diverges

b)

This series converges to $\frac{2}{15}$

c)

This series converges but we cannot find the sum

d)

This series converges to $\frac{2}{25}$

5.

Given the telescoping series $\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+2}\right)$ , find its sum (if possible).

a)

Diverges

b)

$\frac{3}{2}$

c)

$\frac{1}{2}$

d)

This converges, but we cannot find the sum

6.

Which of the following series diverge?

a)

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

b)

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

c)

$\sum_{n=1}^{\infty}\frac{n^6}{n^7}$

d)

$\sum_{n=1}^{\infty}\frac{n^7}{n^6}$

7.

Determine the convergence/divergence of the series

3-\frac{9}{2}+\frac{27}{4}-\frac{81}{8}+...

and find the sum (if possible).

a)

This series diverges

b)

This series converges to $\frac{6}{5}$

c)

This series converges but we cannot find the sum

d)

This series converges to 6

8.

How many terms of the series $\sum_{n=1}^{\infty}\left(-1\right)^{n+1}\frac{1}{n^4}$ are needed to approximate the sum of the series with an error less than 0.001?

a)

3

b)

4

c)

5

d)

6

9.

If the series $\sum_{n=1}^{\infty}\frac{1}{n^{1.01}}$ converges, which of the following series also converges by the direct comparison test?

a)

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

b)

$\sum_{n=1}^{\infty}\frac{1}{n^{1.01}-1}$

c)

$\sum_{n=1}^{\infty}\frac{1}{n^{0.99}}$

d)

$\sum_{n=1}^{\infty}\frac{1}{n^{1.01}+1}$

10.

Find the sum of the telescoping series $\sum_{n=0}^{\infty}\left(\frac{1}{2n+1}-\frac{1}{2n+3}\right)$ .

a)

Diverges

b)

Converges but we can't find the sum

c)

1

d)

$\frac{2}{3}$

11.

Which of the following series is absolutely convergent?

a)

$\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n+1}}{n^{0.99}}$

b)

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

c)

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

d)

$\sum_{n=1}^{\infty}\left(-1\right)^{n+1}3^n$

12.

Which of the following series is conditionally convergent?

a)

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

b)

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

c)

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

d)

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

13.

$\sum_{n=0}^{\infty}\left(\frac{2}{5}\right)^n=$

a)

$\frac{2}{5}$

b)

$\frac{2}{3}$

c)

1

d)

$\frac{5}{3}$

e)

$\frac{5}{2}$

14.

Which of the following series diverges?

a)

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

b)

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

c)

$\sum_{n=1}^{\infty}\frac{0.5}{n}$

d)

$\sum_{n=1}^{\infty}\frac{10^6}{n^3}$

e)

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

15.

Given $\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n+1}}{n^p}$ , what values of p will make this series converge?

a)

$0<p\le1$

b)

$p>0$

c)

$0<p<1$

d)

$p\ge0$

16.

Given $\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n+1}}{n^p}$ , find the maximum value of the error if k terms are used to approximate the sum of the series.

a)

$\frac{1}{p^k}$

b)

$\frac{1}{k^p}$

c)

$\frac{1}{p^{k+1}}$

d)

$\frac{1}{\left(k+1\right)^p}$

17.

Use your calculator to find the 10th partial sum of $\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n+1}}{n^n}$

a)

0.783

b)

1.291

c)

0.951

d)

1.053

18.

Find the sum of the series $\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n+1}}{n^2}$ correct to 0.001 of the true sum.

a)

0.823

b)

0.822

c)

0.821

d)

0.824

19.

Which of the following series converge?

a)

$\sum_{n=1}^{\infty}\frac{8^n}{n!}$

b)

$\sum_{n=1}^{\infty}\frac{n!}{n^{100}}$

c)

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

20.

What is the sum of the series $\sum_{n=1}^{\infty}\frac{\left(-2\right)^n}{e^{n+1}}$

a)

$-\frac{2}{e^2-2e}$

b)

$-\frac{2}{e^2+2e}$

c)

$-\frac{2}{e+2}$

d)

$\frac{e}{e+2}$

e)

This series diverges

21.

The infinite series $\sum_{n=1}^{\infty}a_n$ has nth partial sum $S_n=\left(-1\right)^{n+1}$ for $n\ge1$ . What is the sum of this series?

a)

-1

b)

0

c)

$\frac{1}{2}$

d)

1

e)

The series diverges

22.

The infinite series $\sum_{n=1}^{\infty}a_n$ has nth partial sum $S_n=\frac{n}{3n+1}$ for $n\ge1$ . What is the sum of this series?

a)

$\frac{1}{3}$

b)

1

c)

$\frac{1}{2}$

d)

$\frac{3}{2}$

e)

The series diverges

23.

Consider the series $\sum_{k=1}^{\infty}a_k$ . If $a_1=16$ and $\frac{a_{k+1}}{a_k}=\frac{1}{2}$ for all integers $n\ge1$ , then $\sum_{k=1}^{\infty}a_k$ is...

a)

0

b)

2

c)

17

d)

32

e)

divergent

24.

What are all the values of p for which the series $\sum_{n=1}^{\infty}\frac{1}{n^{2p}+n}$ diverges?

a)

$p\le\frac{1}{2}$

b)

$p<\frac{1}{2}$

c)

$p\ge\frac{1}{2}$

d)

$p>\frac{1}{2}$

e)

The series diverges for all p.

25.

Which of the following series is/are conditionally convergent?

a)

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

b)

$\sum_{k=1}^{\infty}\frac{\left(-1\right)^k}{k^3}$

c)

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

26.

Which of these tests can be used to show the convergence/divergence of $\sum_{n=1}^{\infty}\frac{2^n}{3n}$ ?

a)

ratio test

b)

nth term test

c)

comparison test

d)

geometric series test

e)

p-series test

27.

Which of these tests can be used to show the convergence/divergence of $\sum_{n=1}^{\infty}\frac{3^{2n}}{\left(n+1\right)!}$ ?

a)

ratio test

b)

nth term test

c)

comparison test

d)

geometric series test

e)

integral test

28.

Which of these tests can be used to show the convergence/divergence of $\sum_{n=1}^{\infty}n^{-2}$ ?

a)

p-series test

b)

nth term test

c)

comparison test

d)

geometric series test

e)

integral test

29.

Which of these tests can be used to show the convergence/divergence of $\sum_{n=1}^{\infty}\frac{1}{1+n^2}$ ?

a)

p-series test

b)

ratio test

c)

direct comparison test

d)

limit comparison test

e)

integral test

30.

$\sum_{n=1}^{\infty}\left(-1\right)^{n-1}\frac{8}{\sqrt{n}}$

a)

Converges Absolutely: $\left|a_n\right|$ converges by p-series

b)

Converges Absolutely: $\left|a_n\right|$ diverges by p-series

c)

Converges Conditionally: $\left|a_n\right|$ converges by p-series

d)

Converges Conditionally: $\left|a_n\right|$ diverges by p-series

e)

Diverges by nth term test

31.

$\sum_{n=1}^{\infty}\left(-1\right)^{n-1}\frac{2n^{\frac{1}{3}}}{n^{\frac{5}{3}}}$

a)

Converges Absolutely: $\left|a_n\right|$ converges by p-series

b)

Converges Absolutely: $\left|a_n\right|$ diverges by p-series

c)

Converges Conditionally: $\left|a_n\right|$ converges by p-series

d)

Converges Conditionally: $\left|a_n\right|$ diverges by p-series

e)

Diverges by nth term test

32.

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

a)

Converges Absolutely: $\left|a_n\right|$ converges by geo series

b)

Converges Absolutely: $\left|a_n\right|$ diverges by geo series

c)

Converges Conditionally: $\left|a_n\right|$ converges by geo series

d)

Converges Conditionally: $\left|a_n\right|$ diverges by geo series

e)

Diverges by nth term test

33.

$\sum_{n=1}^{\infty}\left(-1\right)^{n-1}\frac{5n^3+6n}{n^5-1}$

a)

Converges Absolutely: $\left|a_n\right|$ converges by comparison

b)

Converges Absolutely: $\left|a_n\right|$ diverges by comparison

c)

Converges Conditionally: $\left|a_n\right|$ converges by comparison

d)

Converges Conditionally: $\left|a_n\right|$ diverges by comparison

e)

Diverges by nth term test

34.

$\sum_{n=1}^{\infty}\left(-1\right)^{n-1}\frac{3\sqrt{n}}{n^{\frac{3}{2}}-9}$

a)

Converges Absolutely: $\left|a_n\right|$ diverges by comparison

b)

Converges Absolutely: $\left|a_n\right|$ diverges by p-series

c)

Converges Conditionally: $\left|a_n\right|$ diverges by comparison

d)

Converges Conditionally: $\left|a_n\right|$ diverges by p-series

e)

Diverges by nth term test

35.

Which of these series converge conditionally?

a)

$\sum_{n=0}^{\infty}\frac{\left(-1\right)^n}{2n-3}$

b)

$\sum_{n=0}^{\infty}\frac{\cos\left(\pi n\right)}{3^n}$

c)

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

d)

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

36.

Using the Alternating Series Error Bound, what is the minimum number of terms required to guarantee an error less than or equal to 0.01 for the series

\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n+1}}{n^2+4n}

?

a)

9

b)

7

c)

8

d)

10

37.

$\sum_{n=1}^{\infty}\left(-1\right)^n\left(\frac{1}{n^2}+\frac{1}{n^5}\right)$

a)

Absolutely Convergent

b)

Conditionally Convergent

c)

Divergent

38.

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

a)

Absolutely Convergent

b)

Conditionally Convergent

c)

Divergent

39.

Which of the following series are conditionally convergent?

a)

$\sum_{n=0}^{\infty}\frac{\left(-1\right)^n}{n+1}$

b)

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

c)

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

40.

How many terms are needed to get within 0.001 of the true sum of $\sum_{n=0}^{\infty}\frac{\left(-1\right)^n}{\left(2n\right)!}$ ?

a)

4

b)

5

c)

6

d)

3

41.

Does the sequence $a_n=\cos\left(\frac{n}{2}\right)$ converge or diverge?

a)

Diverge

b)

Converge and the limit is 1/2

c)

Converge and the limit is 1

d)

Converge and the limit is 0

42.

Does the sequence $a_n=\cos\left(\frac{2}{n}\right)$ converge or diverge?

a)

Diverge

b)

Converge the limit is 0

c)

Converge the limit is 2

d)

Converge the limit is 1

43.

Does the sequence $a_n=2+(0.8)^n$ converge or diverge?

a)

Diverge

b)

Converge and the limit is 0

c)

Converge and the limit is 2

d)

Converge and the limit is 7

44.

Does the sequence $a_n=\frac{(2n-1)!}{(2n+1)!}$ converge or diverge?

a)

Diverge

b)

Converge and the limit is 1

c)

Converge and the limit is 0

d)

Converge and the limit is 1/4

45.

Does the sequence $a_n=\frac{n^n}{n!}$ converge or diverge?

a)

Diverge

b)

Converge and the limit is 1

c)

Converge and the limit is 0

d)

Converge and the limit is 4/3

46.

Does the sequence $a_n=\ln(n+1)-\ln\left(n\right)$ converge or diverge? (tricky... use some algebra)

a)

Diverge

b)

Converge and the limit is 0

c)

Converge and the limit is 1

d)

Converge and the limit is e

47.

Which of these series converges?

a)

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

b)

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

c)

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

d)

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

48.

Consider the geometric series $\sum_{n=1}^{\infty}a_n$ where $a_n>0$ for all n. The second term of the series is $a_2=162$ and the fourth term is $a_4=18$ . Which of the following statements about $\sum_{n=1}^{\infty}a_n$ is true?

a)

$\sum_{n=1}^{\infty}a_n=486$

b)

$\sum_{n=1}^{\infty}a_n=729$

c)

$\sum_{n=1}^{\infty}a_n=\frac{729}{2}$

d)

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

49.

Which of the following series can be used with the limit comparison test to determine whether the series $\sum_{n=1}^{\infty}\left(\frac{3n^4}{n^5+2}\right)$ converges or diverges?

a)

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

b)

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

c)

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

d)

$\sum_{n=1}^{\infty}\frac{1}{n^4+2}$

50.

Which of the following statements about the series $\sum_{n=0}^{\infty}\frac{2}{n^2}$ is true?

a)

The series converges by the nth term test

b)

The series diverges by the integral test

c)

The series diverges by comparison to the series $\sum_{n=0}^{\infty}\frac{1}{n^2}$

d)

The series converges by comparison to the series $\sum_{n=0}^{\infty}\frac{1}{n^2}$

51.

Which of these series converge conditionally?

a)

$\sum_{n=0}^{\infty}\frac{\left(-1\right)^n}{2n-3}$

b)

$\sum_{n=0}^{\infty}\frac{\cos\left(\pi n\right)}{3^n}$

c)

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

d)

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

52.

Using the Alternating Series Error Bound, what is the minimum number of terms required to guarantee an error less than or equal to 0.01 for the series

\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n+1}}{n^2+4n}

?

a)

9

b)

7

c)

8

d)

10

53.

$\sum_{n=1}^{\infty}\left(-1\right)^n\left(\frac{1}{n^2}+\frac{1}{n^5}\right)$

a)

Absolutely Convergent

b)

Conditionally Convergent

c)

Divergent

54.

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

a)

Absolutely Convergent

b)

Conditionally Convergent

c)

Divergent

55.

Which of the following series are conditionally convergent?

a)

$\sum_{n=0}^{\infty}\frac{\left(-1\right)^n}{n+1}$

b)

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

c)

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

56.

How many terms are needed to get within 0.001 of the true sum of $\sum_{n=0}^{\infty}\frac{\left(-1\right)^n}{\left(2n\right)!}$ ?

a)

4

b)

5

c)

6

d)

3

Unit 10A Review

Total questions: 56

Worksheet time: 3hrs 48mins

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