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

Q1. Which of the following conditions docs NOT ensure the convergence of a real sequence {an}?

(a) |an-an+1| → 0 as n → ∞
(b) Σ|an-an+1| Convergent
(c) Σ nan is convergent
(d) The sequence {a2n}, {a2n+1} and {a3n} are convergent.

Q2. The series Σ {1/2n-1 + (-1)n-1} is

(a) Convergent
(b) divergent
(c) oscillating infinitely
(d) oscíillating finitely

Q3.The sum of the series 1/1! + (1+2)/2! + (1+2+3)/3! + … equals

(a) e
(b) e/2
(c) 3e/2
(d) 1 + e/2

Q4. Let {xn} be a sequence of real numbers so that Σ |xn-x| = c, with c finite. Then

(a) {xn} may not be bounded
(b) {xn} must converge to x
(c) {xn} must converge to x+c
(d) {xn} is bounded but not necessarily convergent.

Q5. Consider the series Σ 1/√n and Σ 1/n√n . Then

(a) both the series converge to the same value.
(b) both the series converge to the different values.
(c) both series are divergent
(d) first series is divergent and second series is convergent

Q6. Which of the following series is divergent?

(a) Σ √n sin(1/n)
(b) Σ √n E sin-1(1/n2)
(c) Σ n tan(1/n3)
(d) Σ tan-1(1/n2)

Q7. The largest interval in which the series Σxn converges

(a) For x with -1 < x < 1
(b) For x with  –1/2 < x < 1/2
(c) For x with -2 < x < 2
(d) For with –1/2 ≤ x ≤ 1/2

Q8. If p is a real number, then the series 1/1p + 1/3p + 1/5p + … to ∞ is convergent for

(a) p > 0
(b) p > 1
(c) p > 2
(d) p < 1

Q9. The series 1/1.4 + 1/2.5 + 1/3.6 + … to ∞ is

(a) Convergent
(b) Divergent
(c) Oscillating
(d) Converges conditionally

Q10. If  Σan is series of positive & negative term and Σqn, Σpn are series of negative and positive terms respectively and if Σan is conditionally convergent then

(a) Σpn is convergent but Σqn not
(b) Σpn is divergent but Σqn not
(c) Σpn & Σqn both are convergent
(d) Σpn & Σqn both are divergent

Q11. Let <an> and <bn> be two sequences of real number such that an = bn – bn+1 for n ∈ N . Then

(a) Convergence of Σan implies the convergence of <bn> not conversely
(b) Convergence of Σan implies the convergence of Σbn not conversely
(c) Convergence of Σan  implies and implied by the convergence of <bn>
(d) Convergence of Σbn  implies and implied by the convergence of <an>

Q12. The alternating Σ(-1)n-1 un converges if {un} is a/an

(a) Decreasing sequence which converges to zero.
(b) Increasing sequence which converges to zero.
(c) Monotone sequence which converges to zero.
(d) Strictly monotone sequence which converges to zero.

Q13. Which one of the following series is convergent ?

(a) 1/2√2 + 1/3√3 + 1/4√4 + … to ∞
(b) 11/2-11/3+11/4-11/5 +… to ∞
(c) 1/21/3 + 1/41/5 +… to ∞
(d) x + x2 + x3 + x4 +… to ∞, where |x|<1

Q14. Select the incorrect statements

(a) Finite set has infinite number of limit points
(b) The sequence 1 + r + r2 + r3 + … + rn, where -1 ≤ r < 1 and n ∈ N is bounded above but not bounded below.
(c) If un > 0 ∀ n then Σun & Σ1/un converge or diverge together
(d) Set of real number except the integers which are multiples of 2 is uncountable set.

Q15. If Σan be a convergent series and an > 0, ∀n ∈ N then the correct statement :

(a) Σ√(an an+1) is convergent
(b) Σan2 is convergent
(c) Σ√(an)/n is convergent
(d) None of the above

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