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**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

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**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 {a _{n}}?**

(a) |a_{n}-a_{n+1}| → 0 as n → ∞

(b) Σ|a_{n}-a_{n+1}| Convergent

(c) Σ na_{n} is convergent

(d) The sequence {a_{2n}}, {a_{2n+1}} and {a_{3n}} are convergent.

**Q2. The series Σ { ^{1}/2^{n-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 {x _{n}} be a sequence of real numbers so that Σ |x_{n}-x| = c, with c finite. Then**

(a) {x_{n}} may not be bounded

(b) {x_{n}} must converge to x

(c) {x_{n}} must converge to x+c

(d) {x_{n}} 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/n^{2})

(c) Σ n tan(1/n^{3})

(d) Σ tan^{-1}(1/n^{2})

**Q7. The largest interval in which the series Σx _{n }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}/1^{p} + ^{1}/3^{p} + ^{1}/5^{p} + … 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 Σa _{n} is series of positive & negative term and Σq_{n}, Σp_{n} are series of negative and positive terms respectively and if Σa_{n} is conditionally convergent then**

(a) Σp_{n} is convergent but Σq_{n} not

(b) Σp_{n} is divergent but Σq_{n} not

(c) Σp_{n} & Σq_{n} both are convergent

(d) Σp_{n} & Σq_{n} both are divergent

**Q11. Let <a _{n}> and <b_{n}> be two sequences of real number such that a_{n} = b_{n} – b_{n+1} for n ∈ N . Then**

(a) Convergence of Σa_{n} implies the convergence of <b_{n}> not conversely

(b) Convergence of Σa_{n} implies the convergence of Σb_{n} not conversely

(c) Convergence of Σa_{n} implies and implied by the convergence of <b_{n}>

(d) Convergence of Σb_{n} implies and implied by the convergence of <a_{n}>

**Q12. The alternating Σ(-1) ^{n-1} un converges if {u_{n}} 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) 1^{1}/_{2}-1^{1}/_{3}+1^{1}/_{4}-1^{1}/_{5} +… to ∞

(c) ^{1}/_{2} – ^{1}/_{3} + ^{1}/_{4} – ^{1}/_{5} +… to ∞

(d) x + x^{2} + x^{3} + x^{4} +… to ∞, where |x|<1

**Q14. Select the incorrect statements**

(a) Finite set has infinite number of limit points

(b) The sequence 1 + r + r^{2} + r^{3} + … + r^{n}, where -1 ≤ r < 1 and n ∈ N is bounded above but not bounded below.

(c) If u_{n} > 0 ∀ n then Σu_{n} & Σ^{1}/u_{n} converge or diverge together

(d) Set of real number except the integers which are multiples of 2 is uncountable set.

**Q15. If Σa _{n} be a convergent series and a_{n} > 0, ∀n ∈ N then the correct statement :**

(a) Σ√(a_{n} a_{n+1}) is convergent

(b) Σa_{n}^{2} is convergent

(c) Σ√(a_{n})/n is convergent

(d) None of the above

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*Answer Key*

*Answer Key*

01. | (a) | 06. | (a) | 11. | (c) |

02. | (d) | 07. | (a) | 12. | (a), (b), (c), (d) |

03. | (c) | 08. | (b) | 13. | (a), (c), (d) |

04. | (b) | 09. | (a) | 14. | (a), (b), (c) |

05. | (d) | 10. | (d) | 15. | (a), (b), (c) |

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