VU Midterm Past Papers
72 solved midterm past paper MCQs for MTH621 (Real Analysis - I) at Virtual University, each with the correct answer marked. Use them to learn the VU question style and test your recall. An independent study tool — not affiliated with VU.
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Q1. A divergent infinite series that does not diverge to +∞ is said to be — — — —
Q2. Which statement(s) is(are) true about the following sequence s0 = 1 and sn = 1 - e^{-n}?
Q3. Every Cauchy sequence has a
Q4. If {sn} is bounded above and lim sup sn = s, then there exists a subsequence {snk} such that lim snk = s
Q5. If {zn} unbounded below, then {zn} has a subsequence {znk} such that lim znk = -∞
Q6. If {zn} unbounded above, then {zn} has a subsequence {znk} such that lim znk = ∞
Q7. If Σ an converges, then lim an = 0
Q8. A sequence {sn} of real numbers is called a Cauchy sequence if for every ε > 0, there is an integer N such that |sn - sm| < ε for all n,m ≥ N
Q9. Suppose that an > 0, bn > 0 and — — — — —. Then Σ bn = ∞ if Σ an = ∞.
Q10. Identify the countable set
Q11. If {an} is an infinite sequence of real numbers, the symbol — — — — — — — — is an infinite series and is the n-th term of series.
Q12. The number e is — — — —
Q13. If Σ an —————, then lim an = 0
Q14. The set A = {x ∈ N : x^2 - 2x + 1 = 0} is equal to
Q15. Which one is an irrational number
Q16. (In Comparison test) Suppose that 0 < an < bn, n ≥ k. Then if Σ an < ∞, Σ bn < — — —.
Q17. Which one is true for the two sets given by T = {x ∈ (0,1) : x is rational} and V = {x ∈ (0,1) : x is irrational}?
Q18. The series Σ an —————, if (-1)^n an > 0, |an+1| < |an|, and lim an = 0.
Q19. The set of terms of {sn_k} is contained in the set of terms of {sn} implies
Q20. Every sequence {sn} of real numbers has a unique limit superior, s*, and a unique limit inferior, s_*, in the extended reals, and — — — —
Q21. Which statement is true about the sequence sn = (-1)^n n?
Q22. If sn = (2n+1)/(n+1), then lim sn = — — — —
Q23. Let Σ an = A and Σ bn = B, where A and B are finite. Then Σ c an = ————
Q24. Identify the identical sequences
Q25. Which one is true for the two sets given by S = {x ∈ R : 0 < x < 1} and T = {x ∈ (0,1) : x is rational}?
Q26. A sequence {sn} of real numbers is called a Cauchy sequence if for every ε > 0, there is an integer N such that |sn - sm| < ε if m,n — — —
Q27. The sum of a convergent series is — — — —
Q28. Suppose that an > 0 and bn > 0 for n ≥ k. Then Σ an = ∞ if Σ bn = ∞ and lim inf an/bn — — —
Q29. If {sn} is — — — —, then lim sn = inf {sn}.
Q30. The statement(s) true about the real number √10 is(are)
Q31. If the sequence is convergent then
Q32. If {sn} is a Cauchy sequence of real numbers, then {sn} is
Q33. If {sn} is ______, then {sn} has a convergent subsequence.
Q34. Cauchy sequence has a
Q35. Which statement(s) is(are) true about the following sequence s1=1 and sn=1−e−n+1?
Q36. ∑an converges, if (−1)nan>0, |an+1|<|an|, and lim an=______.
Q37. If sn<b for any sequence {sn}, where b is a real number, then {sn} is
Q38. If the sequence is increasing, then it
Q39. {sn} converges to −1 and {tn} converges to 5, then the sequence {sn+tn}
Q40. The limit of the sequence 1/n
Q41. If a sequence {sn} is nonincreasing, then
Q42. If {sn} is a sequence of real numbers, then lim sn=s iff lim sup sn = lim inf sn = s.
Q43. A sequence {sn} of real numbers has a unique limit superior, s*, and a unique limit inferior, s*, in the extended reals, and
Q44. If lim sn = +∞, then
Q45. A sequence {sn} of real numbers is called a Cauchy sequence if for every ε>0, there is an integer N such that |sn−sm|<ε if m,n>N.
Q46. For the convergent sequence {sn}, which statement is true?
Q47. If {sn} is bounded above and does not diverge to −∞, then there is a unique real number s such that ∀ε>0, sn<s+ε ______
Q48. If p>0 and a is real, then value of lim n^p / a^n is
Q49. The given series ∑ 1/n^2 =
Q50. If {an} is an infinite sequence of real numbers, the symbol ∑an = ______ and an is the nth term of series.
Q51. If lim sn=A and lim tn=B, where A and B are finite. Then lim (sn+tn) =
Q52. The value of lim 1/n^p is
Q53. lim inf sn = −∞ if
Q54. lim nfs, =—oo lf
Q55. If a sequnce {s,} is nonincreasing, then — — — ——
Q56. If |8u| < r for any sequence {8}, where r is a real number. then {sq} is _
Q57. the sequence is increasing, then it
Q58. limsup s,, = oo if
Q59. Theseries 3° r*, —1<r<1,is
Q60. Apoimt Tis a limit point of a sel iff there is a sequence {z, } of point in § such that ty Zz forn=1, and
Q61. If {x,} unbounded below, then {x,} has a subsequence {z,} such that lim a, = — — ——
Q62. If |z| < 1, then
Q63. Which statement is true about the sequence 5, = (—1)"n
Q64. If {sn}is — — — ——, then lim s, = sup {s,}
Q65. If {zn} is ______, then {zn} has a convergent subsequence.
Q66. A Cauchy sequence has a
Q67. Which statement(s) is(are) true about the following sequence s1 = 1 and sn = 1 - e^(-n) + e^(-2n)?
Q68. For harmonic series ∑ 1/n,
Q69. The limit of the sequence 1/n is
Q70. If p > 0 and a is real, then the value of lim (n^p / (1+n^p)) is
Q71. The given series ∑ 1/n^2 is
Q72. If ∑ an = A and ∑ bn = B, where A and B are finite. Then ∑ (an + bn) =