# NBHM (National Board for Higher Mathematics) MSc and MA Mathematics: Questions 18 - 25 of 101

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## Question number: 18

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Appeared in Year: 2015

### Write in Short

Find the Radius of Convergence of the series

## Question number: 19

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Appeared in Year: 2015

Essay Question▾

### Describe in Detail

Let be the polygon in the complex plane with vertices situated at the nth Roots of unity. Let be the area and let be its perimeter. Evaluate .

### Explanation

The points represented by nth roots of unity are located at the vertices of a regular polygon of n sides, inscribed in a unit circle having center at the origin, one vertex being on the positive real axis (in figure below)

But when n approaches to infinity, the n sided polygon will approach to circle of unit radius.

… (72 more words) …

## Question number: 20

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Appeared in Year: 2008

Essay Question▾

### Describe in Detail

Determine if each of the following series is absolutely convergent, conditionally convergent or divergent:

a)

b)

c)

### Explanation

Option (a) The given series is

Here

Take

By comparison test behave alike

They converge or diverge together

converges by p –test

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## Question number: 21

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Appeared in Year: 2005

Essay Question▾

### Describe in Detail

Let n be a positive integer. Let and let . For what value of n will f be twice differentiable but its second derivative discontinuous at ?

### Explanation

Let

i. e

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## Question number: 22

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Appeared in Year: 2007

MCQ▾

### Question

Let . Which of the following function are analytic in the entire complex plane?

### Choices

Choice (4)Response

a.

b.

c.

d.

Question does not provide sufficient data or is vague

## Question number: 23

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Appeared in Year: 2015

Essay Question▾

### Describe in Detail

Find the sum of the following series upto n terms.

### Explanation

Let be the rth term of the given series. Then

Where

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## Question number: 24

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Appeared in Year: 2014

Essay Question▾

Let . Evaluate

### Explanation

applying

Applying

Taking common from

By

Applying

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## Question number: 25

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Appeared in Year: 2007