# IIT JAM Mathematics Test Series 2023 : Real Analysis – Sequences of Real Numbers

Practice IIT JAM Mathematics Test Series for Free only On www.examflame.com and take your preparation to the another level. In the Series of Tests for IIT JAM 2023 this is the test of Sequences of Real Numbers of the Chapter Real Analysis. These Questions are prepared as per the Latest Syllabus of IIT JAM Mathematics 2023 . Practicing mock tests/Test series, you will get an idea about how and which type of Question will ask in the Examination. It also boost your confidence level. Also Solve IIT JAM Mathematics Previous Year Question Paper. And Don’t Forget to Shore with Your Friends.

## IIT JAM Mathematics Test Series 2023 : Real Analysis – Sequences of Real Numbers

Q1. The sequence an = (A2n2 + n +1)1/2 -n is

(a) convergent for all real values of A
(b) divergent for all real values of A
(c) convergent for exactly one real value of A
(d) Convergent for exactly two real values of A

Q2. The least upper bound of the sequence <1-1/n> is

(a) 0
(b) -1
(c) 1
(d) None of these

Q3. Let {an} and {bn} be sequences of real numbers defined as a1 =1 and for n ≥1, an+1 = an+(-1)n2-n, bn = (2an+1 – an)/an. Then

(a) {an} converges to zero and {bn} is a Cauchy sequence
(b) {an} converges to non-zero and {bn} is a Cauchy sequence
(c) {an} converges to zero and {bn} is not a convergent sequence
(d) {an} converges to non-zero and {bn} is not a convergent sequence

Q4. Which of the following séquence <an> is monotonic:

(a) <an> = 1- (-1)n/n
(b) an+2 = 1/2(an+an+1) where a1<a2 and a1, a2 are given.
(c) an+1 = [(ab2 + an2)/(a+1)]1/2  ∀n , where a>0, 0<a1<b and a1 = a
(d) None of these

Q5. Let {xn} be a real sequence. If sequence of even terms of {xn} converges to 1 and sequence of odd terms converges to -1. Then the sequence {xn} will

(a) Converge to 0
(b) Converge to 1
(c) Converge to -I
(d) None of these

Q6. Consider he interval (1, -1) and a sequence {an}n=1 of elements in it. Then

(a) Every limit point of {an} is in (-1,1)
(b) Every limit point of {an} is in [-1,1]
(c) The limit points of {an} can only be in {-1,0,1}
(d) The limit points of {an} cannot be in {-1,0,1}

Q7. Consider the sequence 4,0,4.1,0,4.11,0, 4.111,0,…  Then

(a) converges to 41/9
(b) is divergent
(c) is unbounded
(d) is not convergent and has supremum = 41/9

Q8. Let p(x) be a polynomial in the real variable x of degree 5. Then limit n→∞ of the function p(n)/2n is

(a) 5
(b) 1
(c) 0
(d) ∞

Q9. If 0<c<d, then the sequence an = (cn +dn)1/n

(a) is bounded and monotonically decreasing
(b) is bounded and monotonically increasing
(c) is monotonically increasing but unbounded for 1<c<d
(d) is monotonically decreasing, but unbounded for 1<c<d

Q10. The limit superior and the limit inferior of the following sequence   <(-1)n(1+1/n)> are

(a) 1, -1
(b) 2, -1
(c) 2, 1
(d) 1, 1

Q11. Let {an}, {bn}and {cn}be sequences of real numbers such that bn=a2n and cn = a2n+1 . Then an is convergent

(a) implies {bn} is convergent but , {cn} need not be convergent
(b) implies {cn} is convergent but , {bn} need not be convergent
(c) implies both {bn} and {cn} are convergent
(d) if both {bn} and {cn} are convergent

Q12. For each positive integer n, let an be the number of points of intersection of the graph y = sinx with the line y = x/n. The sequence {an} is

(a) decreasing
(b) constant
(c) converging to zero
(d) diverging to infinity

Q13. Let <xn> be a convergent sequence and <yn> be a monotonic sequence. Then <xn.yn>

(a) Always converges
(b) Converges if <xn> is monotonic
(c) Converges if <xnyn> is monotonic
(d) Converges if <yn> is bounded

Q14. Let {xn} be an unbounded sequence of non- zero real numbers. Then

(a) {xn} must have a convergent subsequence
(b) {xn} cannot have a convergent subsequence
(c) {1/xn} must have a convergent subsequence
(d) {1/xn}cannot have a convergent Subsequence.

Q15. xn+1 = -3/4 xn , x0=1 . The sequence {xn}

(a) diverges
(b) xn is monotonically increasing and converges to 0
(c) xn is monotonically decreasing and converges to 0
(d) none of the above

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