Real Sequences

Practice Assignment 2
Sagar Surya

(−1)n−1
Q.1 - The sequence ⟨sn ⟩, where sn = converges to
n

(a) 1

(b) -1

(c) 0

(d) None of these

1 1 1 1
Q.2 - lim n [1 + 2 + 3 + ⋯ + n] is __.
n→∞

∞
Q.3 – Let S = {x ∈ ℝ: x ≥ 0, ∑ x √n < ∞} . Then the supremum of S is __.
n=1

1 1 1
Q.4 – Let a1 = 1 and an = an−1 + 4, n ≥ 2. Then, lim [a +a +⋯+ a ] is equal to
n→∞ 1 a2 2 a3 n−1 an
___.
 1
Q.5 - Consider the sequencean = (1 + (−1)n n)n . Then

(a) lim sup an = lim inf an = 1

n→∞ n→∞

(b) lim sup an = lim inf an = e

n→∞ n→∞

1
(c) lim sup an = lim inf an = e
n→∞ n→∞

1
(d) lim sup an = e, lim inf an = e
n→∞ n→∞

1 1 1 1
Q.6 - lim ( + +⋯+ ) = ____.
n→∞ √n √1+√3 √3+√5 √2n−1+√2n+1

Q.7 - Consider the interval [−1,1] and a sequence {an }∞

n=1 of elements in it. Then,

(a) Every limit point of {an }∞

n=1 is in (−1,1)

(b) Every limit point of {an }∞

n=1 is in [−1,1]

(c) The limit points of {an }∞

n=1 is in {−1,0,1}

(d) The limit points of {an }∞

n=1 cannot be in {−1,0,1}
 xn−1 +xn−2
Q.8 - Let x1 = 0, x2 = 1 and for n ≥ 3, define xn = . Which of the following is/
2
are true?

(a) {xn } is monotone sequence

1
(b) lim xn = 2
n→∞

(c) {xn } is a Cauchy sequence

(d) {xn } is a Cauchy sequence

Q.9 - Let the sequence {xn }∞

n=1 be defined by x1 = √2 and xn+1 = (√2)
xn
for n ≥ 1.

Then which one of the following is/are TRUE ?

(a) The sequence {xn } is monotonically increasing and lim xn = 2

n→∞

(b) The sequence {xn } is neither monotonically increasing nor monotonically decreasing

(c) lim xn does not exist

n→∞

(d) lim xn = ∞
n→∞

1 1
Q.10 - For n ≥ 1, the sequence {xn }∞
n=1 , where xn = 1 + +⋯+ − 2√n is
√ 2 √n

(a) Decreasing

(b) Monotone

(c) Increasing

(d) Constant

Q.11 -
For a sequence {an } of real numbers which of the following is a negation of the statement
 lim an = 0 ?
n→∞

(a) There exists ε > 0 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒 𝑠𝑒𝑡 {n ∈ ℕ||an ∣> 𝜀 } is infinite.

(b) For any M > 0, 𝑡ℎ𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡𝑠 𝑁 ∈ ℕ such that |an | > 𝑀 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛 ≥ 𝑁.

(c) There exists a nonzero real number a such that for every ε > 0, 𝑡ℎ𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡𝑠 𝑁 ∈
ℕ with

|an − a| < 𝑀, for all n ≥ N

(d) For any a ∈ ℝ and every ε > 0, 𝑡ℎ𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒𝑙𝑦 𝑚𝑎𝑛𝑦 𝑛 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 |an − a| > 𝜀

Q.12 - Let x1 ∈ (0,1) be real number between 0 and 1. For n > 1, 𝑑𝑒𝑓𝑖𝑛𝑒 xn+1 = xn − xnn+1 .
Then, the

Limit of the sequence exists. True/ False

n2
Q.13 - Let {an } be a sequence of real numbers such that |an+1 − an | ≤ 2n for all n ∈ ℕ. Then

n2
(a) Let {an } be a sequence of real numbers such that |an+1 − an | ≤ for all n ∈ ℕ. Then
2n

(b) The sequence {an } may be unbounded.

(c) The sequence {an } is bounded but may not converge.

(d) The sequence {an } is convergent.

Q.14 - The sequence √7, √7 + √7, √7 + √7 + √7, … converges to ___. ( correct to two
decimals )
 an
Q.15 - Given a1 = 1 and an+1 = an + 1 + a , the value of lim (an+1 − an ) is
n+1 n→∞

(a) 0

(b) 0.5

(c) 1

(d) 2

Q.16 - Which of the following holds/ hold TRUE for the sequence defined as

a0 = 25, an+1 = 2√an ?

(a) The sequence is monotone decreasing

(b) The sequence is monotone increasing

(c) The sequence is Cauchy

(d) The sequence is convergent

Q.17 - Let {xn } be a sequence of positive numbers such that lim xn = x. Define
n→∞

n n−1
1 x x
zn = [x1 (1 (+ )) + √x2 (1 + ) + ⋯ + xn (1 + x)]
n n m−1

Then,

(a) {zn } does not have a limit

(b) {zn } convergs to xe

(c) {zn } convergs to xex

(d) None of these

1 1
Q.18 - lim sin ((2nπ + ) sin (2nπ + )) =___. (correct to two decimals)
n→∞ 2nπ 2nπ
Q.19 –

(a) (xn + yn ) is convergent

(b) (xn + yn ) is bounded

(c) (xn + yn ) has n convergent subsequence

(d) (xn + yn ) has no bounded subrequence

Q.20 - - Let (xn ) and (yn ) be convergent sequences. Then the sequence (3xn 2yn )

(a) Converges

(b) Diverges

(c) May Converge or Diverge

(d) None of these

Q.21 - Let (xn )n≥1 be a sequence of non − negative real numbers. Then which of the
following is TRUE?

(a) lim infxn = 0 ⇒ limxn2 = 0

(b) lim supxn = 0 ⇒ limxn2 = 0

(c) lim infxn = 0 ⇒ (xn )n≥1 is bounded

(d) lim infxn2 > 4 ⇒ 𝑙𝑖𝑚𝑠𝑢𝑝xn > 4

Q.22 - Let (xn )n≥1 be a sequence of non − negative real numbers. Then, which of the
following is TRUE ?

(a) lim infxn = 0 ⇒ limxn2 = 0

(b) lim supxn = 0 ⇒ limxn2 = 0

(c) lim infxn = 0 ⇒ (xn )n≥1 is bounded

(d) lim infxn2 > 4 ⇒ 𝑙𝑖𝑚𝑠𝑢𝑝xn > 4

Q.23 - Let {xn } be a sequence of positive real numbers. Consider a sequence {yn } defined as
x
yn = 1+xn . Then, which of the following is TRUE ?
n

(a) {xn } is convergent if {yn } is convergent.

(b) {yn } is convergent if {xn } is convergent.

(c) {yn } is bounded if {xn } is bounded.

(d) None of these

Q.24 - Let
Q.25 - Let {an } be a sequence of real numbers. Then lim an exists if and only if
n→∞

(a) lim a2n and lim a2n+2 exists

n→∞ n→∞

(b) lim a2n and lim a2n+1 exist

n→∞ n→∞

(c) lim a2n , lim a2n+1 and lim a3n exist

n→∞ n→∞ n→∞

(d) None of these

Q.26 - Let a>0 is fixed. A sequence {𝑎𝑛 } is defined as

𝑎𝑛2 + 3𝑎
𝑎1 > 0 and 𝑎𝑛+1 = 𝑎𝑛 2 for 𝑛 ∈ ℕ
3𝑎𝑛 + 𝑎

Then, which of these is/are True ?

(a) If 0 < 𝑎1 < √𝑎, the sequence is increasing and bounded above by √𝑎

(b) If 𝑎1 > √𝑎, the sequence is decreasing and bounded below by √𝑎

(c) If 𝑎1 = √𝑎, the sequence is constant

(d) The sequence converges to √𝑎

Q.27 - Let a sequence be defined as 𝑎1 ∈ ℝ 𝑎𝑛𝑑

 𝑎𝑛+1 = 𝑎𝑛2 + (1 − 2𝑎)𝑎𝑛 + 𝑎2 𝑓𝑜𝑟 𝑛 ∈ ℕ,

where a is an arbitrarily fixed number. Then the sequence converges if

(a) 𝑎1 > 𝑎 𝑓𝑜𝑟 𝑛 > 1

(b) 𝑎1 < 𝑎 − 1 for n > 1

(c) 𝑎 − 1 ≤ 𝑎𝑛 ≤ 𝑎 𝑓𝑜𝑟 𝑛 > 1

(d) None of these

1
Q.28 - Let ⟨𝑥𝑛 ⟩ = 22𝑛 (1 − 𝑐𝑜𝑠 2𝑛 ) ∀𝑛 ∈ 𝑁, then the sequence ⟨𝑥𝑛 ⟩

(a) Does not converge

(b) Converges to 0

(c) Converges to 0.5

(d) None of these

1 1 1
Q.29 - The sequence defined as 𝑎𝑛 = 3 + 4 𝑚 2 + ⋯ + (𝑛+2) 𝑚 𝑛 , 𝑚 ≥ 2, 𝑚 ∈ ℕ
√ √

(a) Bounded but not convergent

(b) Cauchy but not Convergent

(c) Convergent

(d) Properly Divergent

Q.30 - Consider the following statements.

 1 1 1 1
(I) 𝑙𝑖𝑚 [ + + …… ]=0
𝑛→∞ √ 𝑛 √ 𝑛+1 √ 𝑛+2 √2𝑛

1 1 1
(II) 𝑙𝑖𝑚 ( + + ⋯+ )=0
𝑛→∞ (𝑛+1)2 (𝑛+2)2 (2𝑛)2

Then,

(a) I is TRUE, II is NOT TRUE

(b) II is TRUE, I is NOT TRUE

(c) I is TRUE, II is TRUE

(d) I is FALSE, II is FALSE