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Sequences And Series Of Real Numbers -3 - Mathematics MCQ


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20 Questions MCQ Test Topic-wise Tests & Solved Examples for Mathematics - Sequences And Series Of Real Numbers -3

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Sequences And Series Of Real Numbers -3 - Question 1

If Sn denoted the sum of n term of the series 

Detailed Solution for Sequences And Series Of Real Numbers -3 - Question 1

If Sn denote the sum of n terms of the series  then Sn will be > Explanation:

A: Sn > n^2
- The given comparison tells us that the sum of the first n terms of the series is greater than n^2.
- This could be true for certain series where the sum of terms grows faster than n^2, such as series with exponential growth.
- However, this comparison alone does not provide enough information to determine the behavior of the series for all values of n.

B: Sn > n
- This comparison indicates that the sum of the first n terms of the series is greater than n itself.
- Similar to the first comparison, this could be true for certain series with increasing terms.
- Again, this comparison alone does not give a definitive answer about the convergence or divergence of the series.

C: Sn = ∞
- This comparison suggests that the sum of the terms of the series diverges to infinity.
- If the sum of the series approaches infinity as n increases, it indicates that the series does not converge.
- This comparison implies that the series may be divergent and does not have a finite sum.

D: Sn = ∞
- This comparison reiterates the previous point that the sum of the series diverges to infinity.
- It reinforces the idea that the series does not have a finite sum and diverges as n increases.

Conclusion:
- Based on the comparisons provided, it is most likely that the series described does not converge and has an infinite sum. This is indicated by the comparisons stating that the sum of the terms exceeds n^2 and n, and ultimately diverges to infinity.

Sequences And Series Of Real Numbers -3 - Question 2

If ai > 0 for i = 1, 2, 3,..,n and a1, a2, a3, ...an = 1 then the greatest value of (1 + a1)(1 + a2)... (1 + an) is:

Detailed Solution for Sequences And Series Of Real Numbers -3 - Question 2

Let the given expansion be f(n)

f(n) = (1 + a1)(1 + a2)... (1 + an)

Also given, a1 = a2 = a3 = ... = an = 1

Consider for n = 2

f(2) = (1 + a1)(1 + a2)

f(2) = (1 + 1)(1 + 1) = 22

Consider for n = 5

f(5) = (1 + a1)(1 + a2)(1 + a3)(1 + a4)(1 + a5)

f(5) = (1 + 1)(1 + 1)(1 + 1)(1 + 1)(1 + 1) = 25

Similary for n times, it is given as

f(n) = (1 + 1)(1 + 1) ...... (1 + 1) = 2n

(1 + a1)(1 + a2)... (1 + an) = 2n

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Sequences And Series Of Real Numbers -3 - Question 3

Let < Sn > and <tn> be two sequences such that  and  Then,

Detailed Solution for Sequences And Series Of Real Numbers -3 - Question 3

Since, we know that ifbe a sequence such that  and 


Hence, 
and 

*Multiple options can be correct
Sequences And Series Of Real Numbers -3 - Question 4

The sequence <sn> , where   is

Detailed Solution for Sequences And Series Of Real Numbers -3 - Question 4

If m > n,

Since 2r – 1 ≤ r !

Therefore, it follows that <sn> is a cauchy sequence. Hence <sn> converges
 

Sequences And Series Of Real Numbers -3 - Question 5

Which one of the following is incorrect?

Detailed Solution for Sequences And Series Of Real Numbers -3 - Question 5

The set of all cluster points of a sequence is sometimes called the limit set. contains all but finitely many elements of the sequence). That is why we do not use the term limit point of a sequence as a synonym for accumulation point of the sequence.

Sequences And Series Of Real Numbers -3 - Question 6

For infinite series  for n, and there is a real number N, such that for n ≥ N implies |an|≤ bn.  If  coverges, then

Sequences And Series Of Real Numbers -3 - Question 7

If a sequence  converges to a real number A, then 

Sequences And Series Of Real Numbers -3 - Question 8

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

Detailed Solution for Sequences And Series Of Real Numbers -3 - Question 8

A sequence that is not a Cauchy sequence is called a divergent sequence. A Cauchy sequence is a sequence whose elements get closer and closer together as the sequence progresses. 

Sequences And Series Of Real Numbers -3 - Question 9

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

Sequences And Series Of Real Numbers -3 - Question 10

If a sequence < an2 > converges to a2, then the sequence < an > converges to

Sequences And Series Of Real Numbers -3 - Question 11

A real number l is a limit point of a sequence < an> if and only if there exists

Sequences And Series Of Real Numbers -3 - Question 12

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

Sequences And Series Of Real Numbers -3 - Question 13

Let sequence  converges to A and sequence   converges to B, with an ≤ bn for all n, then

Sequences And Series Of Real Numbers -3 - Question 14

What will be the value of 

Detailed Solution for Sequences And Series Of Real Numbers -3 - Question 14

just divide numertor and denominator by n2 

Sequences And Series Of Real Numbers -3 - Question 15

A : Cauchy sequence is convergent.
B : Cauchy sequence is bounded in R.

Detailed Solution for Sequences And Series Of Real Numbers -3 - Question 15
  • A Cauchy sequence is one where the terms become arbitrarily close to each other as the sequence progresses.
  • In the real numbers (R), every Cauchy sequence converges to a limit. This means A is true.
  • A Cauchy sequence is also bounded, as the terms cannot diverge infinitely far apart. Hence, B is also true.
  • Therefore, both statements A and B are true, making the correct answer option 1.
Sequences And Series Of Real Numbers -3 - Question 16

 is equal to

Sequences And Series Of Real Numbers -3 - Question 17

Let San be a convergent series of positive terms and let Sbn be a divergent series of positive terms. Then,

Detailed Solution for Sequences And Series Of Real Numbers -3 - Question 17

The correct option is 2.

Since San is convergent, its sequence of terms must converge to 0. This is because if the terms did not converge to zero, then the sum would not converge.

On the other hand, Sbn is divergent, which means that its sequence of terms does not converge to 0. If it did, then the series would converge by the nth term test.

Therefore, option 2 is the correct answer.

Sequences And Series Of Real Numbers -3 - Question 18

Which of the following statement is true? 

Detailed Solution for Sequences And Series Of Real Numbers -3 - Question 18

For any positive number ε there is a natural number n, such that

Sequences And Series Of Real Numbers -3 - Question 19

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

Detailed Solution for Sequences And Series Of Real Numbers -3 - Question 19

For C, suppose lim a2n = p, lim a2n+1 = q, lim a3n = r.

Considering a6n, p = r.

Considering a6n+3 = a2(3n+1)+1, q = r.

Therefore p=q=r.

Since a2n and a2n+1 include all integers, limits exists.

So C is true.

Sequences And Series Of Real Numbers -3 - Question 20

Let < an> ≥ a. Let for every positive integer k, Ak be the set of all positive integer n such that | an- a | ≤ 1/k. Then,

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