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HOLY CROSS COLLEGE (Autonomous) TIRUCHIRAPPALLI - 620 002

II B.Sc. MATHEMATICS, III SEMESTER , NOVEMBER, 2014MAJOR CORE 4: SEQUENCES AND SERIES

SUBJECT CODE : U12MA3MCT04

Unit : IIILevel: K

Sub Unit : 3.1Type: MCQ

1. A series un containing positive and negative terms is conditionally convergent if a) un is convergent and un is divergent b) un is convergent c) un is convergent d) un is convergent and un is divergent

KEY: a) un is convergent and un is divergent

2. The series u1-u2+u3-u4+.. is convergent if a)un+1>un n and =0 b)un >un+1 n and c)un >un+1 n and =0 d)un+1 > un n and KEY: c)un >un+1 n and =0

Unit : IIILevel: U

Sub Unit : 3.1Type: MCQ

1.isa) absolutely convergentb) conditionally convergentc) not convergentd) divergent

KEY: b) conditionally convergent

2.is a) conditionally convergent b) absolutely convergentc) convergent d) divergent KEY: d) divergent

3.a) absolutely convergentb) not absolutely convergent

c) conditionally convergent d) divergent

KEY: a) absolutely convergentUnit : III Level: U

Sub Unit : 3.1Type: A & R

1. Of the following pair of statements, one is assertion and the second is a possible reason Read the statements carefully and mark your answer according to the following key.

Assertion :The series is absolutely convergent

Reason :The series is convergent.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 but the reason does not explain the assertion c) The assertion is true but the reason is falsed) Both the assertion and reason are false

KEY: a)The assertion and reason are true statements and the reason is an adequate explanation for the assertion. 2. Of the following pair of statements, one is assertion and the second is a possible reason Read the statements carefully and mark your answer according to the following key.

Assertion :The series is divergent.

Reason :The series is divergent.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 but the reason does not explain the assertion c) The assertion is true but the reason is false d) Both the assertion and reason are falseKEY: d) Both the assertion and reason are false

3. Of the following pair of statements, one is assertion and the second is a possible reason. Read the statements carefully and mark your answer according to the following key.

Assertion:The series is convergent

Reason :The series is convergent.a)The assertion and reason are true statements and the reason is an adequateexplanation for the assertion.b) The assertion and reason are true statements but the reason does not explain the assertionc) The assertion is true but the reason is falsed) Both the assertion and reason are falseKEY: a)The assertion and reason are true statements and the reason is an adequate explanation for the assertion. 4. Of the following pair of statements, one is assertion and the second is a possible reason. Read the statements carefully and mark your answer according to the following key.

Assertion :The series is convergent

Reason :The series is divergent.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 but the reason does not explain the assertion c) The assertion is true but the reason is false d) Both the assertion and reason are falseKEY: c) The assertion is true but the reason is false

5.Of the following pair of statements, one is assertion and the second is a possible reason Read the statements carefully and mark your answer according to the following key.

Assertion :The series is conditionally convergent

Reason :The series is convergent.

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 but the reason does not explain the assertion c) The assertion is true but the reason is false d) Both the assertion and reason are false

KEY: c) The assertion is true but the reason is false

6. Of the following pair of statements, one is assertion and the second is a possible reason . Read the statements carefully and mark your answer according to the following key.

Assertion :The series is conditionally convergent

Reason :The series is convergent though is divergent.

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 but the reason does not explain the assertion c) The assertion is true but the reason is false d) Both the assertion and reason are falseKEY: c) The assertion is true but the reason is false7. Of the following pair of statements, one is assertion and the second is a possible reason . Read the statements carefully and mark your answer according to the following key.

Assertion :The series is conditionally convergent

Reason :The series is divergent

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 but the reason does not explain the assertion c) The assertion is true but the reason is false d) Both the assertion and reason are false

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

8. Of the following pair of statements, one is assertion and the second is a possible reason Read the statements carefully and mark your answer according to the following key.

Assertion :The series is convergent

Reason :The series is divergent.

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 but the reason does not explain the assertion c) The assertion is true but the reason is false d) Both the assertion and reason are falseKEY: b) The assertion and reason are true statements but the reason does not explain the assertion

9. Of the following pair of statements, one is assertion and the second is a possible reason. Read the statements carefully and mark your answer according to the following key.

Assertion :The series is divergent.

Reason : The series is convergent.

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 but the reason does not explain the assertion c) The assertion is true but the reason is false d) Both the assertion and reason are false

KEY: d) Both the assertion and reason are false

10. Of the following pair of statements, one is assertion and the second is a possible reason. Read the statements carefully and mark your answer according to the following key.

Assertion :The series is conditionally convergent

Reason :The series is convergent though is divergent.

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 but the reason does not explain the assertion c) The assertion is true but the reason is false d) Both the assertion and reason are false

KEY: )The assertion and reason are true statements and the reason is an adequate explanation for the assertion. Unit : IIILevel: K

Sub Unit : 3.1Type: VSA

1.When do you say that a series is absolutely convergent? 2.When do you say that a series is conditionally convergent?3.Give an example of a series which is absolutely convergent.4.Given example of a series which is conditionally convergent.5.State Leibnitzs test for testing convergence of alternating series.

Unit : IIILevel: U

Sub Unit : 3.1Type: VSA

1.What is the nature of the series ?

2.What is the nature of the series ?

3 State any one condition of Leibnitzs test for alternating series, which is satisfied by the series

Unit : IIILevel: K

Sub Unit : 3.1Type: PA

1.Prove that an absolutely convergent series is convergent .

2.If u1 u2 + u3 u4 + ... is a series of terms, alternatively positive and negative and if un > un+1 for all values of n and if , prove that the series is convergent.

Unit : IIILevel: U

Sub Unit : 3.1Type: PA

1.Examine for convergency the series where 0 < x < 1.

2. Examine the convergency of the series

3. Prove that the series is absolutely convergent.

4. Test for absolute convergence the series,

5. Examine the convergency of the series,

Unit : III Level: KLevel: K

Sub Unit : 3.1 Type : E

1.(a) Test for convergency the alternating series (-1)n (7) (b)Prove that an alternating series u1 u2 + u3 u4 + ... is convergent if un > un+1 for all values of n and if if

If (8

Type: E

Unit : III Level: USub Unit : 3.1 Type : ELevel: U

Type: E

1 a)Show that the series, is conditionally convergent.(7)

b)Discuss the convergency of the series when 0 < p 1 (8)

2 a) Examine the convergence of the series, .x>0 (7)

b)Test for convergency ,the series (8)

3 a) Show that the series is absolutely convergent (7)

b) Test for convergency the series, for all values of x (8)

4 a)Examine the convergence of the series (7)

b)Discuss the convergency of the series, (8)

Unit: 3Level: U

Subunit: 3.2Type: MCQ

1.Binomial expansion for (1+3x)n is valid when

a) b) c) |x| 3d) |x| > 3

KEY: a) 2.Binomial expansion for (2 5x)-3 is valid when

a) |x| < 1b) |x| < 5c) d)

KEY: c) 3.Binomial expansion for (1-4x)n is valid when

a) |x| 4b) c) |x| > 4d)

KEY: b) 4.The sum to of the series 1+x+x2+x3+ is a) (1+x)-1b) (1-x)-1c) (1+x)-2d) (1-x)-2 KEY: b) (1-x)-15.The sum to of the series 1 2x + 3x2 4x3 + isa) (1+x)-2b) (1-x)-2c) (1+x)-3d) (1-x)-3 KEY: a) (1+x)-26.In the Binomial expansion the successive factors of the numerator are ina) Arithmetic progressionb) geometric progressionc) Harmonic progressiond) Arithmetic geometric progression. KEY: a ) Arithmetic progression

Unit: 3Level: K

Subunit: 3.2Type: VSA

1.State Binomial theorem for a rational index?2.For what values of x is the binomial expansion (1+x)n valid?3.Give the expansion for (1-x)-14.Give the expansion for (1+x)-15.Give the expansion of (1-x)-26.Write the expansion of (1+x)-2

Unit: 3Level: U

Subunit: 3.2Type: PA

1.Prove that 1 + n.

=

2.Show that + = 0

3.Show that the series

4.Find that sum to infinity of the series,

5.Sum to infinity the series

6.Sum to infinity the series

7.Sum the series to infinity of the series

8.Find the sum to infinity of the series

9.Find the sum to infinity

Unit: 3Level: K

Subunit: 3.2Type: E

1. State and prove Binomial Theorem for a rational index .

Unit: 3Level: U

Subunit: 3.2Type: E

1. Find the sum of (15)

2. a)Sum the series to infinity, (7) b)Sum the series to infinity (8)

3. a) Sum the series to infinity (7)

b) Sum the series to infinity (8)

Unit: 3Level: U

Subunit: 3.3Type: MCQ

1.When x is small the value of is approximately a) 1- 3x b) 1 + 3xc) 1 + 2xd) 1 2xKEY: d) 1 2x

2.When x is small, the value of (1 4x)-3/4 is approximately a) 1 + 3xb) 1 3x c) 1 + 4xd) 1 3xKEY: a) 1 + 3x

3.When x is small, is

a) b) c) d)

KEY: d) 4.When x is large, (x3+1)1/3 is

a) b) c) 1 x3d)

KEY: a) 5.When x is large, (x3+6)1/3 is

a) b) c) d)

KEY: b) Unit: 3Level: U

Subunit: 3.3Type: VSA

1.Find the value of 2.Find the value of (998)1/3 correct to 3 decimal places

3.Find the value of correct to 5 places of decimals.

4.Find the value of correct to 4 decimal places.

5.Find the value of correct to 4 decimal places.Unit: 3Level: U

Subunit: 3.3Type: PA

1.If x is so small that x3 and higher powers of x can be neglected, show that the nth root of (1+x) equals nearly.2.If x be so small that its square and higher powers may be neglected, find the value of (1-7x)1/3. (1+2x)-3/4.

3.If p-q is small compared with p or q, show that will be nearly equal to 4.If x is small, prove that

nearly

5.If x is large, show that nearly.

6.If x is large, show that nearly.7.Find the value of (998)1/3 correct to 6 decimal places.8.Find the value of (1003)1/3 (997)1/3 correct to 6 decimal places.

9. Calculate correct to six places of decimals

10. If =a+x when x is very small, then =a. approximately.

Unit: 3Level: U

Subunit: 3.3Type: E

1. a)Show that when x is small, nearly (7) b)Find the value of (1.02)3/2 (0.98)3/2 correct to six places of decimals. (8)

2. a)If x is large, show that nearly. (7) b)Find to five places of decimals, the value of (1002)1/3 (998)1/3(8)

3.a)Find the value of (7) b) When x is small, prove that approximately (8)

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