Unit -I

HOLY CROSS COLLEGE (Autonomous) TIRUCHIRAPPALLI - 620 002II B.Sc. MATHEMATICS, SEM III, AUGUST 2014MAJOR CORE 4: SEQUENCES AND SERIES

Subject Code : U12MA3MCT04

Unit : ILevel: K

Sub Unit : 1.1Type: MCQ

1.If { an } and { bn } are convergent sequences, converging to a and b respectively, then { an + bn } is a) oscillating between a and bb) converging to a + bc) converging to abd) diverging

Key:(b)converging to a + b

2.If { an } and { bn } are convergent sequences, converging to a and b respectively, then { an - bn } isa) may diverge to -b)may diverge to +c) converges to a - b d) converges to abKey:(c) converges to a b

3.If { an } converges to a and { bn } converges to b0, then converges to

a) a + bb) a bc) abd)

Key:(d) 4.If { an } converges to a and { bn } converges to b, then { anbn } converges to

a) b) ab c) d) a2 b2 Key:(b)ab

5.If { an } converges l, then tends to a) lb) l c) 0d) + Key:(a)l

Unit : I

Level: U


1.is

a) converging to b) converging to

c) diverging to -d) oscillating between and

Key:(d) oscillating between and 2.{2 + (-1)n} isa) converging to 3b) converging to 1c) oscillating between 1 and 3 d) diverging to + Key:(c) oscillating between 1 and 3

3.{ n2} tends toa) 0b) +c) -d) 1 Key:(b) +

4.If an = (-1)n.n, then { an } isa) an oscillating sequenceb) a convergent sequencec) a divergent sequence d) a bounded sequence Key:(a) an oscillating sequence

5.If {an} = {n} and {bn} = {-n}, then {an+bn} is a) converging to 0b) diverging to +c) diverging to - d) oscillating between - and +

Key:(a) converging to 0

6.If {an} = and {bn} = {n2}, then {an+bn} a) converges b) divergesc) oscillatesd) bounded Key:(b)diverges

7.If an = and bn = n, then {anbn}a) converges to 1b) diverges to +c) oscillates between 0 and d) tends to 0 Key:(a) converges to 1

8.If an = n2 and bn = -n2, then tends to a) 1b) -1c) +d) - Key:(b) -1

9.If {an} = and {bn} = {n2}, then {anbn} tends toa) a definite limitb) +c) - d) two different limits Key:(b) +

10.If {an} = and {bn} = {n2}, then tends toa) 0b) +c) -d) 1 Key:(a) 0

11.If an = n, bn = -n2, then {an + bn}a) tends to 0b) diverges to -c) diverges to +d) oscillates between - and +

Key:(b) diverges to -

12.a) tends to the limit 0b) oscillates between - and +c) tends to -d) oscillates between 0 and Key:(d) oscillates between 0 and

13. a) convergesb) diverges to +c) oscillatesd) diverges to - Key:(c) oscillates

14.The sequnce a) oscillatesb)converges to 0c) diverges to + d) diverges to - Key:(b) converges to 0

15.The sequnce isa) oscillatoryb)divergentc) convergent d) unboundedKey:(c) convergent

16.

a) 1 b) c) d) 0

Key:(b)

17.If an= n, then is a) 0b) 1c) -1d) +Key:(a) 0

Unit : ILevel: U

Sub Unit : 1.1Type: M&C

1.Match the given sequences with their converging natures and choose the correct answer:

1.The sequence A) oscillates infinitely 2. The sequence {-n}B) is a constant sequence3.The sequence {1+(-1)n} C) diverges4.The sequence n{1+(-1)n.2} D) oscillates finitely E) converges

a) 1E, 2C, 3D, 4Ab) 1D, 2E, 3A, 4Bc) 1E, 2C, 3B, 4Ad) 1B, 2C, 3D, 4A

Key:(a) 1E, 2C, 3D, 4A

2.Match the given sequences with their converging natures and choose the correct answer:

1. A) diverges

2. B) converges

3. C) oscillates finitely4. {(-1)n n} D) monotonically increasing E) oscillates infinitely

a) 1D, 2A, 3E, 4Bb) 1C, 2A, 3B, 4Ec) 1C, 2A, 3E, 4Bd) 1D, 2C, 3B, 4E

Key:(b) 1C, 2A, 3B, 4E

3.Match the given sequences with their respective converging natures and choose the correct answer:

1. A) oscillatory

2. B) converges to

3. C) diverges to

4. D) converges to zero E) converges to one

a) 1E, 2A, 3D, 4Cb) 1D, 2A, 3E, 4Cc) 1D, 2A, 3B, 4Cd) 1E, 2C, 3B, 4A Key:(b) 1D, 2A, 3E, 4C

Unit : ILevel: U

Sub Unit : 1.1Type: A&R

1. Of the following pair of statements one is an assertion and the other is a possible reason. Read the statements carefully and choose the correct answer from the key given:

Assertion: is a convergent sequence

Reason:tends to the unique limit 0 as n 1. The assertion and reason are true statements and the reason is an adequate explanation for the assertion.1. The assertion and reason are true statements and the reason does not explain the assertion.1. The assertion is a true statement but the reason is a false statement.1. The assertion and the reason are false statement.

Key: (a) The assertion and reason are true statements and the reason is an adequate explanation for the assertion


Assertion:{(-1)n} is an oscillating sequence.

Reason:{(-1)n} is a bounded sequence.

a) The assertion and reason are true statements and the reason is an adequate explanation for the assertion.b) The assertion and reason are true statements and the reason does not explain the assertion.c) The assertion is a true statement but the reason is a false statement.d) The assertion and the reason are false statement.

Key: (b) The assertion and reason are true statements and the reason does not explain the assertion


Assertion: is a convergent sequence

Reason: is a bounded sequencea) The assertion and reason are true statements and the reason is an adequate explanation for the assertion.b) The assertion and reason are true statements and the reason does not explain the assertion.c) The assertion is a true statement but the reason is a false statement.d) The assertion and the reason are false statement. Key:(b) The assertion and reason are true statements and the reason does not explain the assertion

4. Of the following pair of statements one is an assertion and the other is a possible reason. Read the statements carefully and choose the correct answer from the key given:Assertion : {(-1)n n} converges Reason: {(-1)n n} is bounded


Key:(d) The assertion and the reason are false statement.5. Of the following pair of statements one is an assertion and the other is a possible reason. Read the statements carefully and choose the correct answer from the key given:

Assertion : oscillates

Reason: is unboundeda) The assertion and reason are true statements and the reason is an adequate explanation for the assertion.b) The assertion and reason are true statements and the reason does not explain the assertion.c) The assertion is a true statement but the reason is a false statement.d) The assertion and the reason are false statement.

Key:(d) The assertion and the reason are false statement.6 .Of the following pair of statements one is an assertion and the other is a possible reason. Read the statements carefully and choose the correct answer from the key given:

Assertion : converges

Reason: is unbounded


Key: (c) The assertion is a true statement but the reason is a false statement.

Unit : ILevel: K

Sub Unit : 1.1Type: VSA

1.Define a sequence.2.What is the difference between a set and a sequence?3.Define the limit of a sequence.4.Define a convergent sequence with an example.5.When can you say that a sequence diverges to ?6.Define an oscillating sequence 7.Given an example of a sequence which oscillates finitely.8.Give an example of a sequence which oscillates infinitely.9. Give an example of a sequence which oscillates between 0 and .10. Give an example of a convergent sequence.11. Give an example of a divergent sequence.12. Give an example of an oscillating sequence.13. When can you say that a sequence diverges to -?

Unit : ILevel: U


1.What is the limit of the sequence ?

2.What is the limit of the sequence ?

3 What is the limit of the sequence ?

4. What is the limit of the sequence ?

5. What is the limit of the sequence ?

6. What is the nature of the sequence ?



9. What is the nature of the sequence ?10.What is the nature of the sequence { 2+(-1)n }?

11.What is the nature of the sequence ?






17.What is the nature of the sequence?


Unit : ILevel: U

Sub Unit : 1.1Type: PA

1.If {an} and {bn} are convergent sequences, then prove that {an + bn} is also a convergent sequence.2.If the sequences {an} and {bn} are both convergent, then prove that {an - bn} is also convergent.

3.If {an} converges to a and {bn} converges to b(0),then prove that converges to .

Unit : ILevel: K

Sub Unit : 1.2 Type: MCQ

1.If and ,then

a) b) c) d)

Key:(b) 2.If {an} is not bounded above then

a) b) c)d) Key:(d)

3. If {an} is not bounded below then

a) b) c) d)

Key:(b)

Unit : ILevel: U


1.{-n} isa) bounded aboveb) bounded belowc) bounded both above and belowd) bounded neither above nor below. Key:(a) bounded above

2.{2n} isa) bounded above b) bounded belowc) bounded both below and above d)bounded neither above nor below Key:(b) bounded below

3.is boundeda) above but not below b) below but not abovec) both above and belowd) neither above nor below Key:(c) both above and below

4. is boundeda) above b) belowc) both above and belowd) neither above nor below

Key:(d) neither above nor below

5. isa) 1b) 0 c) +d) -

Key:(b) 0

6. isa) + b) - c) 0d) 2 Key:(a) +

Unit : ILevel: U

Sub Unit : 1.2Type: M&C

1.Match the sequences with their nature of boundedness and choose the correct answer:1.{ n2 } A) bounded both above and below2.{ -n2 } B) bounded neither above nor below

3. C) bounded between 2 and 34. { (-1)nn} D) not bounded aboveE) not bounded below

a) 1E, 2D, 3A, 4Bb) 1D, 2E, 3A, 4Bc) 1C, 2A, 3D, 4E d) 1B, 2C, 3E, 4A Key:(b) 1D, 2E, 3A, 4B

2.Match the sequences with their nature of boundedness and choose the correct answer:

1. A) bounded between -1and 0

2. B) bounded between 3 and 5

3. C) bounded between 0and 14. { 4+(-1)n} D) bounded between 1 and 2

E) bounded between 2 and 3

a) 1C, 2E, 3A, 4Bb) 1C, 2D, 3A, 4Bc) 1C, 2E, 3D, 4B d) 1D, 2E, 3B, 4A

Key:(a) 1C, 2E, 3A, 4B

3.Match the limits with their values and choose the correct answer:

1. A) 0

2. B) +

3. C) -

4. D)-1 E) 1

a) 1E, 2B, 3C, 4Ab) 1E, 2C, 3B, 4Ac) 1D, 2E, 3B, 4A d) 1D, 2C, 3B, 4A

Key:(b) 1E, 2C, 3B, 4A

Unit : ILevel: U

Sub Unit : 1.2Type: AR

1. Of the following pair of statements one is an assertion and the second is a possible reason. Read the statements carefully and choose the correct answer.

Assertion: The inferior numbers do not exist for the sequence {-n2} Reason: The sequence {-n2} is bounded above but not bounded below.

1. The assertion and reason are true statements and the reason gives an adequate explanation for assertion.1. The assertion and reason are true statements but the reason does not explain the assertion.1. The assertion is a true statement but the reason is a false statement.1. The assertion and the reason are false statement.

Key: (a) The assertion and reason are true statements and the reason gives an adequate explanation for assertion.


Assertion: is convergent

Reason: is a Cauchy sequencea) The assertion and reason are true statements and the reason is an adequate explanation for the assertion.b) The assertion and reason are true statements and the reason does not explain the assertion.c) The assertion is a true statement but the reason is a false statement.d) The assertion and the reason are false statement.


3. Of the following pair of statements one is an assertion and the second is a possible reason. Read the statements carefully and choose the correct answer. Assertion: {n } is not convergent Reason:{n} is not a Cauchy sequencea) The assertion and reason are true statements and the reason is an adequate explanation for the assertion.b) The assertion and reason are true statements and the reason does not explain the assertion.c) The assertion is a true statement but the reason is a false statement.d) The assertion and the reason are false statement.



Assertion: Reason: {n} has no inferior numbers.a) The assertion and reason are true statements and the reason is an adequate explanation for the assertion.b) The assertion and reason are true statements and the reason does not explain the assertion.c) The assertion is a true statement but the reason is a false statement.d) The assertion and the reason are false statement. Key: (c) The assertion is a true statement but the reason is a false statement

5. Of the following pair of statements one is an assertion and the second is a possible reason. Read the statements carefully and choose the correct answer. Assertion: {n} has no inferior number Reason:{n} is not bounded belowa) The assertion and reason are true statements and the reason is an adequate explanation for the assertion.b) The assertion and reason are true statements and the reason does not explain the assertion.c) The assertion is a true statement but the reason is a false statement.d) The assertion and the reason are false statement.

Key: (d) The assertion and the reason are false statement

6. Of the following pair of statements one is an assertion and the second is a possible reason. Read the statements carefully and choose the correct answer. Assertion: {-n} has no inferior number Reason:{-n} is not bounded abovea) The assertion and reason are true statements and the reason is an adequate explanation for the assertion.b) The assertion and reason are true statements and the reason does not explain the assertion.c) The assertion is a true statement but the reason is a false statement.d) The assertion and the reason are false statement.

Key: (c) The assertion is a true statement but the reason is a false statement


Assertion: Reason : {n} is bounded belowa) The assertion and reason are true statements and the reason is an adequate explanation for the assertion.b) The assertion and reason are true statements and the reason does not explain the assertion.c) The assertion is a true statement but the reason is a false statement.d) The assertion and the reason are false statement.

Key: (b) The assertion and reason are true statements but the reason does not explain the assertion

Unit : ILevel: K


1.When do you say that a set is bounded above?2.When do you say that a set is bounded below?3.Define the upper limit of a sequence.4. Define the lower limit of a sequence.5.State Cauchys principle of convergence of a sequence.

Unit : I

Level: U


1.Give an example of a sequence for which the upper limit does not exist.2.Give an example of a sequence for which the lower limit does not exist.

Unit : ILevel: U


1.{(-1)n isa) a monotonically increasing sequence b) a monotonically decreasing sequencec) a bounded sequence d) an unbounded sequence

Key:(c) a bounded sequence

2. isa) monotonically decreasingb) monotonically increasingc) tending to the limit 0d) tending to the limit Key:(a) monotonically decreasing

3.The sequence isa) tending to the limit 0b) neither increasing nor decreasingc) monotonically increasingd) monotonically decreasing Key:(c) monotonically increasing

4.The sequence isa) monotonically decreasingb) monotonically increasing c) tending to the limit 0d) tending to the limit 1 Key:(a) monotonically decreasing

5.The sequence isa) monotonically decreasingb) monotonically increasing c) tending to the limit 1 d) tending to the limit Key:(a) monotonically decreasingUnit : ILevel: U

Sub Unit : 1.3Type: A & R

1. Of the following statements, the first is an assertion and the other is a possible reason. Read the statements carefully and choose the correct answer from the key given.

1.Assertion: The sequence {an} where an = is a bounded sequence.

Reason: is bounded between 2 and 3

1. The assertion and reason are true statements and the reason gives an adequate explanation for assertion.1. The assertion and reason are true statements and the reason gives an adequate explanation for assertion.1. The assertion is a true statement but the reason is a false statement.1. The assertion and the reason are false statement.

Key: (a) The assertion and reason are true statements and the reason gives an adequate explanation for assertion

2. Of the following pair of statements one is an assertion and the second is a possible reason. Read the statements carefully and choose the correct answer from the key given:

Assertion: The sequence {an} where an = is a bounded sequence.

Reason: is bounded between 1 and 2(a) The assertion and reason are true statements and the reason gives an adequate explanation for assertion.(b) The assertion and reason are true statements and the reason gives an adequate explanation for assertion.(c) The assertion is a true statement but the reason is a false statement.(d) The assertion and the reason are false statement.


3. Of the following pair of statements one is an assertion and the second is a possible reason. Read the statements carefully and choose the correct answer from the key given:

Assertion: The sequence is monotonically decreasing.

Reason: For the sequence , an+1>an, n(a) The assertion and reason are true statements and the reason gives an adequate explanation for assertion.(b) The assertion and reason are true statements and the reason gives an adequate explanation for assertion.(c) The assertion is a true statement but the reason is a false statement.(d) The assertion and the reason are false statement.


4. Of the following pair of statements one is an assertion and the second is a possible reason. Read the statements carefully and choose the correct answer from the key given:Assertion: The sequence {n+1} is monotonically increasing

Reason : For the sequence {n+1}, an+1>an, n(a) The assertion and reason are true statements and the reason gives an adequate explanation for assertion.(b) The assertion and reason are true statements and the reason gives an adequate explanation for assertion.(c) The assertion is a true statement but the reason is a false statement.(d) The assertion and the reason are false statement.Key: (a) The assertion and reason are true statements and the reason gives an adequate explanation for assertion

5. Of the following pair of statements one is an assertion and the second is a possible reason. Read the statements carefully and choose the correct answer from the key given:Assertion: The sequence {-n} is monotonically increasing

Reason: For the sequence {-n}, an+1>an, n(a) The assertion and reason are true statements and the reason gives an adequate explanation for assertion.(b) The assertion and reason are true statements and the reason gives an adequate explanation for assertion.(c) The assertion is a true statement but the reason is a false statement.(d) The assertion and the reason are false statement.

Key: (d) The assertion and the reason are false statement

6.Of the following pair of statements one is an assertion and the second is a possible reason. Read the statements carefully and choose the correct answer from the key given:

Assertion: The sequence is monotonically increasing

Reason: The sequence is bounded(a) The assertion and reason are true statements and the reason gives an adequate explanation for assertion.(b) The assertion and reason are true statements and the reason gives an adequate explanation for assertion.(c) The assertion is a true statement but the reason is a false statement.(d) The assertion and the reason are false statement.

Key: (b) The assertion and reason are true statements and the reason gives an adequate explanation for assertion

Unit : ILevel: K


1.Define a monotonically increasing sequence. 2.Define a monotonically decreasing sequence.3.Is the sequence {(-1)n} monotonically increasing or not?why?4. Give an example of a monotonically increasing sequence. 5. Give an example of a monotonically decreasing sequence.

Unit : ILevel: U


1.Is the sequence monotonically increasing or decreasing?why?

2.Is a monotonically increasing sequence or not?why?

3.Is a monotonically increasing sequence or not?

4. Is a monotonically decreasing sequence or not?

5. Is a monotonically decreasing sequence or not?

Unit : ILevel: U


1. If an+1= (an+bn),bn+1= , Show that the sequence {an} and {bn} converge to a common limit

2.If an+2 = where an >0 Show that the sequence {a2n-1} and {a2n } are both monotonic,one increasing and the other decreasing and also prove that

{an} tends to

3.Prove that exists and find its limit.

4. If {an} converges to l, then prove that also exists and is equal to l.

5.If {an} converges to a and {bn} converges to b,then prove that {anbn} converges to ab. 6.State and prove Cauchys general principle of convergence.

Unit : ILevel: K


1.Prove that a sequence cannot converge to two different limits 2.Prove that a monotonic sequence always tends to a limit finite or infinite.

3. If an=Show that the sequence {an} tends to a limit. 4. Prove that the necessary and sufficient condition for the sequence {an} to

converge is that

Unit : ILevel: U

Sub Unit : 1.3Type: E

1.a)Prove that exists and find its limit. (7)

b) If {an} converges to l, then prove that also exists and is equal to l. (8)

2.a)If {an} converges to a and {bn} converges to b,then prove that {anbn} converges to ab. (7) b)State and prove Cauchys general principle of convergence. (8)

3.a)Prove that exists and find its limit. (7) b)State and prove Cauchys general principle of convergence. (8)

4.a)If {an} converges to a and {bn} converges to b,then prove that {anbn} converges to ab. (7)

b) If {an} converges to l, then prove that also exists and is equal to l. (8)

Unit : I

Level: K

Sub Unit:1.3Type: E

1.a)Prove that a sequence cannot converge to two different limits(7) b)Prove that a monotonic sequence always tends to a limit finite or infinite.(8)

2. a) If an=Show that the sequence {an} tends to a limit. (7) b) Prove that the necessary and sufficient condition for the sequence {an} to

converge is that (8)3.a)Prove that a sequence cannot converge to two different limits(7) b) Prove that the necessary and sufficient condition for the sequence {an} to

converge is that (8)

4. a) If an=Show that the sequence {an} tends to a limit. (7) b)Prove that a monotonic sequence always tends to a limit finite or infinite. (8)

Unit -I

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