# NBHM (National Board for Higher Mathematics) MSc and MA Mathematics: Questions 52 - 59 of 101

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## Question number: 52

Appeared in Year: 2015

### Describe in Detail

In each of the following cases, state whether the given series is absolutely convergent, conditionally convergent or divergent

a)

b)

c)

### Explanation

Option (a) The given series is

Where

Now comparing it with

1)

2)

3)

By

… (486 more words) …

## Question number: 53

Appeared in Year: 2006

### Question

Pick out series which are absolutely convergent.

### Choices

Choice (4) | Response | |
---|---|---|

a. |
| |

b. |
| |

c. |
| |

d. | Both a. and b. are correct |

## Question number: 54

» Sequences and Series (Convergence)

Appeared in Year: 2005

### Describe in Detail

What are the value of for which the following series is convergent?

### Explanation

Let the omen series be denoted by ……. . eq. (1)

Where

Since for absolute convergence, we take

Now is convergent if …………eq. (2)

And divergent if (By p-test)

Also eq. (1) is convergent for by Leibnitz test

is convergent b

… (66 more words) …

## Question number: 55

» Order of Pole and Its Residue

Appeared in Year: 2010

### Describe in Detail

Find the order of the pole and its residue at of the function

### Explanation

Order of pole

At , order of pole is n if

And , is the smallest such integer.

… (359 more words) …

## Question number: 56

» Line Integral in Complex Plane

Appeared in Year: 2017

### Describe in Detail

Let C be the contour in the complex plane consisting of two straight line segments, one from and the other from . Evaluate

Where

### Explanation

**Along OM**

The imaginary axis from is the line segment

Now on so that

Also y varies from

…………. eq. (1)

**Along MP**

MP is the line parallel to t

… (245 more words) …

## Question number: 57

Appeared in Year: 2007

### Describe in Detail

Evaluate

### Explanation

Let ……. eq. (1)

…………. eq. (2)

Adding (1) and (2), we get

………. eq. (3)

Let

… (441 more words) …

## Question number: 58

» Differentiability in Complex Plane

Appeared in Year: 2008

### Describe in Detail

Which of the following maps are differentiable everywhere?

a)

b) Such that for all

c) Where

### Explanation

Option (a)

Clearly f is continuous for all

Clearly f is differentiable for all except we can’t say at

At

… (621 more words) …

## Question number: 59

Appeared in Year: 2010

### Describe in Detail

In each of the following verify whether the series is absolutely convergent conditionally or divergent:

a)

b)

c)

### Explanation

Option (a) The given series is

Where

Now

…………. eq. (1)

does not converge

Given series is not absolutely convergent

Also it is not convergent by Leibnitz test (by eq. (1) )

The given series diverge

Opti

… (645 more words) …