# NBHM (National Board for Higher Mathematics) MSc and MA Mathematics: Questions 43 - 49 of 101

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## Question number: 43

» Convergence of Positive Series Test

Appeared in Year: 2014

### Describe in Detail

Which of the following series are convergent?

a)

b)

c) , Where

### Explanation

Option (a) The given series is where

Take

… (142 more words) …

## Question number: 44

Appeared in Year: 2016

### Describe in Detail

Let be a fined positive integer. Let

Evaluate

### Explanation

Consider the expansion ………. . eq. (1)

Integrating both sides of (1) within limits, we get

… (117 more words) …

## Question number: 45

Appeared in Year: 2007

### Write in Short

Let be a seventh root of unity. Write down a polynomial equation of degree satisfied by w.

## Question number: 46

» Integration in Complex Plane.

Appeared in Year: 2010

### Describe in Detail

Let C be the semicircle where varies from , in the complex plane. Evaluate:

### Explanation

Equation of given circle is

… (98 more words) …

## Question number: 47

» Residues of Complex Function

Appeared in Year: 2009

### Write in Short

Let

Write down the residues of at each of its poles

## Question number: 48

Appeared in Year: 2007

### Describe in Detail

Let C denote the Boundary of the square whose sides are given by the lines . Assume that C is described in the position sense, i. e. anticlockwise. Evaluate

### Explanation

To evaluate

Take

Here is a pole of order 1 and lies within the square.

Th

… (77 more words) …

## Question number: 49

Appeared in Year: 2014

### Describe in Detail

Let be differentiable at and let . Evaluate

### Explanation

Now

So this is form (Exponential form)

Now using the result

… (121 more words) …