G12MAN Mathematical Analysis, Revision Quizon First-Year Material

Joel Feinstein

School of Mathematical SciencesUniversity of Nottingham

2009-2010

Joel Feinstein (University of Nottingham) G12MAN Revision Quiz 2009-2010 1 / 7

About this quiz

This quiz does not count towards the assessment of the module. Itis only intended to help you to identify areas which you might wishto ask questions about or to revise further.

You will not be required to reveal your score on this quiz!Answers will be discussed at the end of the quiz, and you willhave the opportunity to ask questions then.Model solutions to these questions, with full justification, willbe made available from the module web page.

Joel Feinstein (University of Nottingham) G12MAN Revision Quiz 2009-2010 2 / 7

About this quiz

This quiz does not count towards the assessment of the module. Itis only intended to help you to identify areas which you might wishto ask questions about or to revise further.You will not be required to reveal your score on this quiz!

Answers will be discussed at the end of the quiz, and you willhave the opportunity to ask questions then.Model solutions to these questions, with full justification, willbe made available from the module web page.

Joel Feinstein (University of Nottingham) G12MAN Revision Quiz 2009-2010 2 / 7

About this quiz

This quiz does not count towards the assessment of the module. Itis only intended to help you to identify areas which you might wishto ask questions about or to revise further.You will not be required to reveal your score on this quiz!Answers will be discussed at the end of the quiz, and you willhave the opportunity to ask questions then.

Model solutions to these questions, with full justification, willbe made available from the module web page.

Joel Feinstein (University of Nottingham) G12MAN Revision Quiz 2009-2010 2 / 7

About this quiz

This quiz does not count towards the assessment of the module. Itis only intended to help you to identify areas which you might wishto ask questions about or to revise further.You will not be required to reveal your score on this quiz!Answers will be discussed at the end of the quiz, and you willhave the opportunity to ask questions then.Model solutions to these questions, with full justification, willbe made available from the module web page.

Joel Feinstein (University of Nottingham) G12MAN Revision Quiz 2009-2010 2 / 7

About this quiz

In this quiz you should simply write down the answers you believeto be correct: no additional justification is required.

For the rest of this module, the default requirement is that youmust justify your answers with an argument sufficient toconvince Dr Feinstein.Such arguments may sometimes be fairly concise, as long as theyinclude the key points.

Joel Feinstein (University of Nottingham) G12MAN Revision Quiz 2009-2010 3 / 7

About this quiz

In this quiz you should simply write down the answers you believeto be correct: no additional justification is required.For the rest of this module, the default requirement is that youmust justify your answers with an argument sufficient toconvince Dr Feinstein.

Such arguments may sometimes be fairly concise, as long as theyinclude the key points.

Joel Feinstein (University of Nottingham) G12MAN Revision Quiz 2009-2010 3 / 7

About this quiz

In this quiz you should simply write down the answers you believeto be correct: no additional justification is required.For the rest of this module, the default requirement is that youmust justify your answers with an argument sufficient toconvince Dr Feinstein.Such arguments may sometimes be fairly concise, as long as theyinclude the key points.

Joel Feinstein (University of Nottingham) G12MAN Revision Quiz 2009-2010 3 / 7

Question 1

Question 1What is the difference between a sequence and a series of realnumbers?

Joel Feinstein (University of Nottingham) G12MAN Revision Quiz 2009-2010 4 / 7

Question 2

Question 2For each of the following sequences (xn), determine whether thesequence converges or diverges.Also, for each sequence which converges, write down the limit of thesequence.

(a) xn = 1/n ;

(b) xn =n2 + 2n + 32n2 + n + 1

;

(c) xn = (n + 1)2 − n2;(d) xn =

√n + 1−

√n ;

(e) xn = (1 + 1n )

n ;

(f) xn =n!

(100)n .

Joel Feinstein (University of Nottingham) G12MAN Revision Quiz 2009-2010 5 / 7

Question 2

Question 2For each of the following sequences (xn), determine whether thesequence converges or diverges.Also, for each sequence which converges, write down the limit of thesequence.

(a) xn = 1/n ;

(b) xn =n2 + 2n + 32n2 + n + 1

;

(c) xn = (n + 1)2 − n2;(d) xn =

√n + 1−

√n ;

(e) xn = (1 + 1n )

n ;

(f) xn =n!

(100)n .

Joel Feinstein (University of Nottingham) G12MAN Revision Quiz 2009-2010 5 / 7

Question 2

Question 2For each of the following sequences (xn), determine whether thesequence converges or diverges.Also, for each sequence which converges, write down the limit of thesequence.

(a) xn = 1/n ;

(b) xn =n2 + 2n + 32n2 + n + 1

;

(c) xn = (n + 1)2 − n2;(d) xn =

√n + 1−

√n ;

(e) xn = (1 + 1n )

n ;

(f) xn =n!

(100)n .

Joel Feinstein (University of Nottingham) G12MAN Revision Quiz 2009-2010 5 / 7

Question 2

Question 2For each of the following sequences (xn), determine whether thesequence converges or diverges.Also, for each sequence which converges, write down the limit of thesequence.

(a) xn = 1/n ;

(b) xn =n2 + 2n + 32n2 + n + 1

;

(c) xn = (n + 1)2 − n2;

(d) xn =√

n + 1−√

n ;(e) xn = (1 + 1

n )n ;

(f) xn =n!

(100)n .

Joel Feinstein (University of Nottingham) G12MAN Revision Quiz 2009-2010 5 / 7

Question 2

Question 2For each of the following sequences (xn), determine whether thesequence converges or diverges.Also, for each sequence which converges, write down the limit of thesequence.

(a) xn = 1/n ;

(b) xn =n2 + 2n + 32n2 + n + 1

;

(c) xn = (n + 1)2 − n2;(d) xn =

√n + 1−

√n ;

(e) xn = (1 + 1n )

n ;

(f) xn =n!

(100)n .

Joel Feinstein (University of Nottingham) G12MAN Revision Quiz 2009-2010 5 / 7

Question 2

Question 2For each of the following sequences (xn), determine whether thesequence converges or diverges.Also, for each sequence which converges, write down the limit of thesequence.

(a) xn = 1/n ;

(b) xn =n2 + 2n + 32n2 + n + 1

;

(c) xn = (n + 1)2 − n2;(d) xn =

√n + 1−

√n ;

(e) xn = (1 + 1n )

n ;

(f) xn =n!

(100)n .

Joel Feinstein (University of Nottingham) G12MAN Revision Quiz 2009-2010 5 / 7

Question 2

Question 2For each of the following sequences (xn), determine whether thesequence converges or diverges.Also, for each sequence which converges, write down the limit of thesequence.

(a) xn = 1/n ;

(b) xn =n2 + 2n + 32n2 + n + 1

;

(c) xn = (n + 1)2 − n2;(d) xn =

√n + 1−

√n ;

(e) xn = (1 + 1n )

n ;

(f) xn =n!

(100)n .

Joel Feinstein (University of Nottingham) G12MAN Revision Quiz 2009-2010 5 / 7

Question 3

Question 3For each of the following series, determine whether the seriesconverges or diverges, and write down the name of the test that youwould use.

(a)∞∑

n=1

(−1)n

2n + 3;

(b)∞∑

n=1

23n + 4

;

(c)∞∑

n=1

sin nn3 + 1

;

(d)∞∑

n=1

3n

n!;

(e)∞∑

n=1

(−1)n

2 + 1n

.

Joel Feinstein (University of Nottingham) G12MAN Revision Quiz 2009-2010 6 / 7

Question 3

Question 3For each of the following series, determine whether the seriesconverges or diverges, and write down the name of the test that youwould use.

(a)∞∑

n=1

(−1)n

2n + 3;

(b)∞∑

n=1

23n + 4

;

(c)∞∑

n=1

sin nn3 + 1

;

(d)∞∑

n=1

3n

n!;

(e)∞∑

n=1

(−1)n

2 + 1n

.

Joel Feinstein (University of Nottingham) G12MAN Revision Quiz 2009-2010 6 / 7

Question 3

Question 3For each of the following series, determine whether the seriesconverges or diverges, and write down the name of the test that youwould use.

(a)∞∑

n=1

(−1)n

2n + 3;

(b)∞∑

n=1

23n + 4

;

(c)∞∑

n=1

sin nn3 + 1

;

(d)∞∑

n=1

3n

n!;

(e)∞∑

n=1

(−1)n

2 + 1n

.

Joel Feinstein (University of Nottingham) G12MAN Revision Quiz 2009-2010 6 / 7

Question 3

Question 3For each of the following series, determine whether the seriesconverges or diverges, and write down the name of the test that youwould use.

(a)∞∑

n=1

(−1)n

2n + 3;

(b)∞∑

n=1

23n + 4

;

(c)∞∑

n=1

sin nn3 + 1

;

(d)∞∑

n=1

3n

n!;

(e)∞∑

n=1

(−1)n

2 + 1n

.

Joel Feinstein (University of Nottingham) G12MAN Revision Quiz 2009-2010 6 / 7

Question 3

Question 3For each of the following series, determine whether the seriesconverges or diverges, and write down the name of the test that youwould use.

(a)∞∑

n=1

(−1)n

2n + 3;

(b)∞∑

n=1

23n + 4

;

(c)∞∑

n=1

sin nn3 + 1

;

(d)∞∑

n=1

3n

n!;

(e)∞∑

n=1

(−1)n

2 + 1n

.

Joel Feinstein (University of Nottingham) G12MAN Revision Quiz 2009-2010 6 / 7

Question 3

Question 3For each of the following series, determine whether the seriesconverges or diverges, and write down the name of the test that youwould use.

(a)∞∑

n=1

(−1)n

2n + 3;

(b)∞∑

n=1

23n + 4

;

(c)∞∑

n=1

sin nn3 + 1

;

(d)∞∑

n=1

3n

n!;

(e)∞∑

n=1

(−1)n

2 + 1n

.

Joel Feinstein (University of Nottingham) G12MAN Revision Quiz 2009-2010 6 / 7

Question 4

Question 4

Question 4For each of the following functions from ℝ to ℝ, determine whetherthey are continuous or discontinuous.

(a) f (x) =

{cos x if x ≥ 0,x + 1 otherwise.

(b) g(x) =

{1 if x ∕= 0,0 if x = 0.

(c) h(x) =

⎧⎨⎩sin x

xif x ∕= 0,

1 if x = 0.

Joel Feinstein (University of Nottingham) G12MAN Revision Quiz 2009-2010 7 / 7

Question 4

Question 4

Question 4For each of the following functions from ℝ to ℝ, determine whetherthey are continuous or discontinuous.

(a) f (x) =

{cos x if x ≥ 0,x + 1 otherwise.

(b) g(x) =

{1 if x ∕= 0,0 if x = 0.

(c) h(x) =

⎧⎨⎩sin x

xif x ∕= 0,

1 if x = 0.

Joel Feinstein (University of Nottingham) G12MAN Revision Quiz 2009-2010 7 / 7

Question 4

Question 4

Question 4For each of the following functions from ℝ to ℝ, determine whetherthey are continuous or discontinuous.

(a) f (x) =

{cos x if x ≥ 0,x + 1 otherwise.

(b) g(x) =

{1 if x ∕= 0,0 if x = 0.

(c) h(x) =

⎧⎨⎩sin x

xif x ∕= 0,

1 if x = 0.

Joel Feinstein (University of Nottingham) G12MAN Revision Quiz 2009-2010 7 / 7

Question 4

Question 4

Question 4For each of the following functions from ℝ to ℝ, determine whetherthey are continuous or discontinuous.

(a) f (x) =

{cos x if x ≥ 0,x + 1 otherwise.

(b) g(x) =

{1 if x ∕= 0,0 if x = 0.

(c) h(x) =

⎧⎨⎩sin x

xif x ∕= 0,

1 if x = 0.

Joel Feinstein (University of Nottingham) G12MAN Revision Quiz 2009-2010 7 / 7