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Sequences And Series Of Real Numbers -7 - Mathematics MCQ


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20 Questions MCQ Test - Sequences And Series Of Real Numbers -7

Sequences And Series Of Real Numbers -7 for Mathematics 2024 is part of Mathematics preparation. The Sequences And Series Of Real Numbers -7 questions and answers have been prepared according to the Mathematics exam syllabus.The Sequences And Series Of Real Numbers -7 MCQs are made for Mathematics 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Sequences And Series Of Real Numbers -7 below.
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Sequences And Series Of Real Numbers -7 - Question 1

 is a sequence of real numbers satisfying   then

Sequences And Series Of Real Numbers -7 - Question 2

If x1 = 4 and   then the sequence {xn} converge to

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Sequences And Series Of Real Numbers -7 - Question 3

  and an = bn - bn + 1 then the series ∑an

Sequences And Series Of Real Numbers -7 - Question 4

The sequence {xn}, where xn  converge to

Sequences And Series Of Real Numbers -7 - Question 5

Let (an) be an increasing sequence of positive real numbers such that the series is divergent. Let  for n = 2, 3, .... Then tn is equal to

Detailed Solution for Sequences And Series Of Real Numbers -7 - Question 5


Sequences And Series Of Real Numbers -7 - Question 6

Which of the following series is divergent?

Detailed Solution for Sequences And Series Of Real Numbers -7 - Question 6

The series Sigma [sin(1/n)] is a divergent series, because

lim(n-->inf.)[sin(1/n)]/(1/n) = 1, which is nonzero, which in turn is a consequence of lim(x-->0)[sin(x)]/x = 1. Hence the two series Sigma [sin(1/n)] and Sigma(1/n) have the same convergence behaviour by limit comparison test, for series of positive terms But we know that the harmonic series Sigma(1/n) diverges. Hence the given series Sigma[sin(1/n) also diverges.

Note that for all 1 </= n, 0 < (1/n) </= 1 < π, and so sin(1/n) is positive i.e. the given series is of positive terms.

Sequences And Series Of Real Numbers -7 - Question 7

  only of

Sequences And Series Of Real Numbers -7 - Question 8

Let ∑un be a series of positive terms such that  Then,
(i) if l > 1, the series converges;
(ii) if l < 1, the series diverges;
(iii) if l = 1, the series may either converge or diverge and therefore the test fails;
​This theorem is known as

Detailed Solution for Sequences And Series Of Real Numbers -7 - Question 8

Correct Answer :- A

Explanation : The series for Rabee’s test is :

Converge when there exists a c>1 such that l is greater than and equal to c for all n>N.

Diverge when l is less than and equal to 1 for all n>N.

Otherwise, the test is inconclusive.

Sequences And Series Of Real Numbers -7 - Question 9

The series 

Sequences And Series Of Real Numbers -7 - Question 10

The set of all x at which the power series converges is

Detailed Solution for Sequences And Series Of Real Numbers -7 - Question 10

Here power series is


 

Sequences And Series Of Real Numbers -7 - Question 11

A series  converges, then sequence 

Sequences And Series Of Real Numbers -7 - Question 12

If n1/n —> 1 as n —> ∞, then the series 

Sequences And Series Of Real Numbers -7 - Question 13

If an > 0 for all n, then is equal to 

Sequences And Series Of Real Numbers -7 - Question 14

If {xn} is a sequence of real numbers, then match list I with list II and select the correct answer using codes given below the lists
List I (sequences)

List II (limit of sequences)

Sequences And Series Of Real Numbers -7 - Question 15

The domain of convergence for 

Detailed Solution for Sequences And Series Of Real Numbers -7 - Question 15

The given series is : x - x2/2 + x3/3 - x4/4.........(A)

The series formed by the coefficients is : 

1 - 1/2 + 1/3 - 1/4.........(B)By I the series is absolutely convergent when x lies between −1 and +1  

By II the series is divergent when x is less than −1 or greater than +1

By III there is no test when x = ±1.  

Thus the given series is said to have [−1, 1] as the interval of convergence.

Sequences And Series Of Real Numbers -7 - Question 16

∑un is a series of positive terms. If Sn = u1 + n2 + ... + un, then

Sequences And Series Of Real Numbers -7 - Question 17

If p > 0, then 

Sequences And Series Of Real Numbers -7 - Question 18

Which amongst the following expressions is not true? 

Sequences And Series Of Real Numbers -7 - Question 19

 where an > 0 n, then (a1 a2 a3...an)1/n = l. Above theorem is known as

Sequences And Series Of Real Numbers -7 - Question 20

Which amongst the following expresses the Cauchy’s first theorem on limits?

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