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QUESTION: 1

If then

Solution:

**Correct Answer :- D**

**Explanation : **For |a|<1

b = 1/a, |b|>1

lim (n→∞) a_{n} = lim (n→∞) (1/b)^{n}

= lim(n→∞) 1/b^{n}

= 1/(±∞)

= 0

QUESTION: 2

Which amongst the following statements is not true?

Solution:

QUESTION: 3

A Cauchy sequence is convergent, if it is a

Solution:

QUESTION: 4

If radius of convergence of series is 1, then radius of convergence of series is…

Solution:

By the differentiation’s theorem, we have

...(i)

Differentiating term by term, we obtain

...(ii)

So, both the series have same radius of convergence.

QUESTION: 5

If {x_{n}} and {y_{n}} are two convergent sequence such that x_{n} < y_{n}, n ∈ N, then

Solution:

**The correct option is Option A.**

**Let lim xn = x and lim yn = y and zn = yn - xn**

** n->∞ n->∞**

**Then (zn) is a convergent sequence such that zn > 0 ∇ n ∊N**

**and lim zn = y - x. **

** n->∞**

**Now (zn) is a convergent sequence of real numbers and zn > 0 ∇ n ∊N**

**So, lim zn ≥ 0**

** n->∞**

**So, y - x ≥ 0**

** => y ≥ x**

** => lim yn ≥ lim xn**

** n->∞ n->∞**

**Hence, proved.**

QUESTION: 6

Let a = min{x^{2} + 2x + 3, x ∈ R} and then the value of

Solution:

QUESTION: 7

The sequence

Solution:

**Correct Answer :- b**

**Explanation : **Suppose otherwise, that there exists a number L implies R and a positive integer N such that

| f(n) - L | < e {for all } e > 0 {for all } n > N.

Since N is a positive integer, we know 4N > N and 4N+2 > N.

But,

f(4N) = cos (2N pi) = 1 and f(4N+2) = cos((2N+1)pi) = -1

Taking e = 1/2

= |1 - L| < 1/2 and |-1-L| < 1/2

=> |1+L| < 1/2

But, these imply

|1-L| + |1+L| < 1.

By the triangle inequality we then have

|1 - L + 1 + L | < 1

=> 2 < 1

a contradiction. Hence, there is no such limit L.

Therefore, the sequence converges to zero.

QUESTION: 8

If for any e > 0, there exists a positive integer m such that I a_{n} - a_{m} | < e whenever n ≥ m . The sequence a_{n} > is called

Solution:

QUESTION: 9

A convergent sequence is a Cauchy sequence, if it is a

Solution:

**ANSWER :- b**

**Solution :- If {an}∞n=1 is a cauchy sequence of real numbers and if there is a sub-sequence of this sequence, {anj}∞j=1 which converges to a real number L, then I need to show that the sequence {an}∞n=1 converges to the real number L.**

QUESTION: 10

Solution:

QUESTION: 11

Solution:

QUESTION: 12

For the given Arithmetic progression find the position of first negative term? 50, 47, 44, 41,............

Solution:

Let n^{th} term=0, the next term would be first negative term.

0 = 50 + (n - 1) – 3, n = 17.66.. therefore at n = 18 the first negative term would occur.

QUESTION: 13

In the given AP series the term at position 11 would be? 5, 8, 11, 14, 17, 20.........50.

Solution:

n^{th} term = a + (n – 1)d, n^{th} term = 5 + (11 - 1)3 = 35.

QUESTION: 14

Which amongst the following statements is not true?

Solution:

QUESTION: 15

If sequences are convergent, then

Solution:

QUESTION: 16

A sequence contains a convergent subsequence, if it is

Solution:

QUESTION: 17

be a sequence converges to 0 and be a sequence that is bounded, then is a sequence that

Solution:

QUESTION: 18

Which of the following sequences of functions is uniformly convergent on (0, 1)?

Solution:

QUESTION: 19

The sequence {x_{n}}, where x_{n} = n^{l/n}, converge to

Solution:

QUESTION: 20

Let x_{n }= 2^{2n} for all n ∈ N. Then the sequence {x_{n}}

Solution:

x_{n} = 2^{2n} for all n ∈ N

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