Unit -IV

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

Subject Code : U12MA3MCT04

Unit : IVLevel: K

Sub Unit : 4.1Type: MCQ

1.e-x is equal to

(a) (b) (c) (d)

KEY : (a)

2. is equal to

(a) (b) e(c) 1(d) 0KEY : (b) e

3. is a (a) binomial series(b) exponential series(c) logarithmic series(d) geometric seriesKEY : (b) exponential series

4.The sum of the series is

(a) (b) (c) (d)

KEY : (c)

5.The sum of the series

(a) (b) (c) (d)

KEY : (c) 6.2x+ + +

(a) (b) (c)(d)

KEY : (b)

Unit : IVLevel: U


1.The co-efficient of xn in e3x is

(a) (b) (c) (d)

KEY : (a) Unit : IVLevel: K

Sub Unit : 4.1Type: M & C

1. Match the following exponential functions with their corresponding formula and choose the correct answer.

1. exA)

2. B)

3. e-xC)

4. D)

E)

(a) 1D, 2C, 3A, 4E(b) 1C, 2D, 3B, 4A(c) 1C, 2E,3A, 4B(d) 1B, 2A, 3D, 4CKEY : (b) 1C, 2D, 3B, 4A

2.Match the following exponential functions with their corresponding formula and choose the correct answer.

1. A)

2. B)

3. eC)

4. e-1D)

E) (a) 1D, 2C, 3B, 4A(b) 1E, 2D, 3A, 4B(c) 1C, 2E, 3B, 4A(d) 1B, 2A, 3D, 4C

KEY : (a) 1D, 2C, 3B, 4A

Unit : IVLevel: K

Sub Unit : 4.1Type: A & R

1.Assertion:e is an incommensurable number.

Reason: e cannot be a fraction of the form .(a) The assertion and the reason are true statements and the reason is an adequate explanation for the assertion.(b) The assertion and the reason are true statements but the reason does not explain the assertion.(c) Assertion is a true statement, but the reason is a false statement.(d) The assertion and the reason are false statements

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

Sub Unit : 4.1Type: VSA

1.What is the value of ?2.What is the value of e?3.State the exponential theorem.4.Write the expression for ax.5.Write the expansion of ex.6.Write the expansion of e-x.

7.Write the series for .

8.Write the expression for .




12.Write the value of .Unit : IVLevel: U


1.Write the nth term of the series .

2.Write the sum of the series .

3.Write the sum of the series .

Unit : IV Level: U

Sub Unit : 4.1Type: PA

1.Show that the co-efficient of xn in the infinite series is .

2.Find the co-efficient of xn in the expansion of .

3.Find the coefficient of xn in the expansion of

4.Find the coefficient of xn in the expansion of (3+2x)

5..Expand as a power series in x and write down the cofficent of xn

6.Show that the coefficient of xr in the expansion of is .

7.Evaluate .

8.Show that the expansion of in powers of x as far as x4 is

9.Show that .

10.S.T.

11.Prove that

12. Show that .

13.Find the sum to infinity of the series



16.Find the sum to infinity of the series 1+

17.Sum to infinity the series 18.Sum to the series + +

19.Sum to the series 20.Sum the series + + +.

Unit : IVLevel: K

Sub Unit : 4.1Type: E

1(a)Prove that the co-efficient of xn in the expansion of is .(7) (b)State and prove exponential theorem.(8)

2 (a)Sum to the series 1+ + + +.(7) (b)Show that = 5e 1(8)

Unit : IVLevel: U


1(a)Sum the series .(7)

(b)Sum the series .(8)

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

(b)Show that = 27 e.(8)


(b)Prove that the infinite series .(8)

4(a)Show that (7)

(b)Show that the sum of the series (8)

5(a)Find the sum to infinity of the series .(7 )

(b)Find the sum to infinity of the series .(8)

6(a)Find the coefficient of xn in the expansion of in powers of x . If the coefficient of x4 is unity and the term containing x10 is missing determine a and b.(7)

(b)S.T the sum to infinity of the series (8)

7(a)Find the values of a,b,c so that the coefficient of xn in the expansion of (a + bx + cx2)ex in ascending powers of x shall be. (7) (b)Show that 5+ + + +.= 13e(8)


(b)Find the sum to infinity of .(8)

9(a)Find the sum to infinity of the series (7) (b)Find the sum to of the series + + + +.(8)

10(a)Find the sum to infinity of the series (7) (b)S.T. = + (8)

11(a).Find the sum to infinity of the series (7) (b). S.T. + . +. + ..=5e(8)

Subject Code : U12MA3MCT04Level: K

Unit : IVType: MCQ

Sub Unit : 4.2

1.Expansion for log(1 + x) is valid for(a) 1 x 1(b) 1 < x 1(c) 0 x 1(d) 1 xKEY : (b) 1 < x 1

2. is equal to

(a) (b) (c) (d)

KEY : (a) 3.Expansion of log (1 x) is

(a) (b) (c) (d)

KEY : (c)

Unit : IVLevel: U


1.The series (2x) is(a) log (1 + x)(b) log (1 x)(c) log (1 2x)(d) log (1 + 2x)KEY : (d) log (1 + 2x)

2.The series is

(a) (b) (c) (d)

KEY : (c)

3.The series is

(a) log (1 + x)(b) (c) (d) log xKEY : (d) log x

Unit : IVLevel: K


Match the following logarithmic functions with their corresponding formula and choose the correct answer.

1.1. log (1 + x)A.

2. log (1 x)B.

3. {log (1 + x)}2C.

4. D.

(a) 1C, 2A, 3D, 4B(b) 1C, 2A, 3B, 4D(c) 1E, 2A, 3D, 4C(d) 1B, 2D, 3E, 4C

KEY : (a) 1C, 2A, 3D, 4B

Unit : IVLevel: K


1.Assertion:log (1 + 2x) can be expanded when |x| < 2Reason:The logarithmic expression is valid, in the range 2 < x < 1.(a) The assertion and reason are true statements and the reason is an adequate explanation for the assertion.(b) Both are true statements, but the reason does not explain the assertion.(c) Assertion is a true statement, but the reason is a false statements.(d) The assertion and the reason are false statements.

KEY : (d) The assertion and the reason are false statements.

Unit : IVLevel: K


1.State the logarithmic series.2.If 1 < x < 1 then expand log (1 + x).3.Write the expansion of log (1 x), when 1 < x < 1.

4.Write the expansion of .Unit : IVLevel: U


1.Write the expansion for log (1 + 3x).2.Write the expansion for log (1 4x).

Unit : IVLevel: U


1.If a, b, c denote three consecutive integers, show that

.

2.Show that ..

3.Sum the series

4..Prove that

5.Show that

6.Show that

7.Show that

8.Show that

9.Prove that =

10.Sum the series (1) + () + ()+.

Unit : IVLevel: U


1(a)Show that .(8)

(b).(7)

2(a)Show that .(7)

(b)Show that .(8)

Unit : IVLevel: K


1.The nth term of the series is

(a) (b) (c) (d)

KEY : (b)

Unit : IVLevel: U


1.The series is equal to

(a) log 2(b) (c) log 3(d) log 1KEY : (a) log 2

2.The sum of is

(a) (b) (c) log n(d) log n + 1

KEY : (a)

Unit : IVLevel: U


Match the given series with their corresponding nth term and choose the correct answer

1.1. A)

2. B)

3. C)

4. D)

(E) (a) 1D, 2A, 3B, 4C(b) 1D, 2A, 3E, 4C(c) 1C, 2A, 3D, 4E(d) 1D, 2C, 3B, 4A

KEY : (a) 1D, 2A, 3B, 4C

Unit : IVLevel: K


1.Assertion:If un = , then as n , un .Reason: is known as Eulers constant.(a) Both are true statements and the reason is adequate explanation for the assertion.(b) Both are true statements but the reason does not explain the assertion.(c) Assertion is true but the reason is false.(d) Both are false statements.

KEY : (a) Both are true statements and the reason is adequate explanation for the assertion.

Unit : IVLevel: K


1.Define Eulers constant.2.Write the expression for log 2.

Unit : IVLevel: U


1. Show that .2. Show that 3. Show that 4. Show that 5. Sum the series .6. Sum to infinity the series whose nth term is 7. Sum the series 8. Sum the series 9. Sum the series

Unit : IVLevel: U


1(a)Sum the series when | x | < 1.(7)

(b)Find the sum of the series .(8)

2(a)Show that (7)

(b)Show that (8)

3(a)Sum to infinity (7)

(b)Sum the series (8)

Unit -IV

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