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QUESTION: 1

If S_{n} denoted the sum of n term of the series

Solution:

If S_{n} denote the sum of n terms of the series then S_{n} will be

QUESTION: 2

Which of the following (s) is/are correct?

Solution:

(given)

QUESTION: 3

Let < S_{n} > and <t_{n}> be two sequences such that and Then,

Solution:

Since, we know that ifbe a sequence such that and

Hence,

and

*Multiple options can be correct

QUESTION: 4

The sequence <s_{n}> , where is

Solution:

If m > n,

Since 2r – 1 ≤ r !

Therefore, it follows that <s_{n}> is a cauchy sequence. Hence <s_{n}> converges

QUESTION: 5

Which one of the following is incorrect?

Solution:

QUESTION: 6

For infinite series for n, and there is a real number N, such that for n ≥ N implies |a_{n}|≤ b_{n}. If coverges, then

Solution:

QUESTION: 7

If a sequence converges to a real number A, then

Solution:

QUESTION: 8

If a sequence is not a Cauchy sequence, then it is a

Solution:

QUESTION: 9

is an increasing bounded sequence, then for the sequence is following statement is false

Solution:

QUESTION: 10

If a sequence < a_{n}^{2} > converges to a^{2}, then the sequence < a_{n} > converges to

Solution:

QUESTION: 11

A real number l is a limit point of a sequence < a_{n}> if and only if there exists

Solution:

QUESTION: 12

be tw0 sequence such that converges respectively to A and AB, then converges Iff

Solution:

QUESTION: 13

Let sequence converges to A and sequence converges to B, with a_{n} ≤ b_{n} for all n, then

Solution:

QUESTION: 14

What will be the value of

Solution:

QUESTION: 15

A : Cauchy sequence is convergent.

B : Cauchy sequence is bounded.

Solution:

QUESTION: 16

is equal to

Solution:

QUESTION: 17

Let Sa_{n} be a convergent series of positive terms and let Sb_{n} be a divergent series of positive terms. Then,

Solution:

QUESTION: 18

Which of the following statement is true? (a) For any positive number ε there is a natural number n, such that . n (ft) Between any two real number there is no irrational number. (c) Convergent sequence is not bounded. (d) None of the above is true.

Solution:

QUESTION: 19

Let {a_{n}} be a sequence of real numbers. Then exists if and only if

Solution:

For C, suppose lima_{2n} = p, lim a_{2n+1 }= q, lim a_{3n }= r.

Considering a_{6n}, p = r.

Considering a_{6n+3 }= a_{2(3n+1)+1}, q = r.

Therefore p=q=r.

Since a_{2n} and a_{2n+1} include all integers, limanliman exists.

So C is true.

QUESTION: 20

Let < a_{n}> —> a. Let for every positive integer k, A_{k} be the set of all positive integer n such that | a_{n}- a | < —1/k. Then,

Solution:

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