8. Which of the following statements are true?Vn? sin(n²)1.I. limn +1n-00II. Let {an} be a convergent sequence of reala1 + ... + annumbers.If bnthennthe sequence {bn} is also convergent andlim b, = lim ɑn.n-00n00III. limarctan n: 0.n-004nIV. Lemma: For any sequence {an} of real numbers,lim a2kk00= lim a2k+1 = L _if and only iflim an = L; (L E R).n-00By the lemma above, one can obtain that1jf an(-1)"-1n23+n41then lim |an|n00(a)I, III(b)III, IV(c)II, IV(d)I, II, III(e)II, III, IV

As per the question we are given three different mathematical claims which we have to verify and…

Which of the following statements are true? ³n² sin(n²) I. lim : 1. n→∞0 n+1 II. Let {an} be a convergent sequence of real a₁ + ... + an numbers. If b = then n the sequence {n} is also convergent and lim b= lim an. n-x n-x arctan n III. lim 0. n-x 4n IV. Lemma: For any sequence {an} of real numbers, lim a2k = lim a2k+1 = L if and only if k→∞ k→∞ lim an = L; (LER). 14x By the lemma above, one can obtain that 1 2 (−1)n-1n if an + n n then lim an = 14x (a) I, III (b) III, IV (c) II, IV (d) - I, II, III (e) II, III, IV SIN n + 48 315 112 n n

Which of the following statements are true? ³n² sin(n²) I. lim : 1. n→∞0 n+1 II. Let {an} be a convergent sequence of real a₁ + ... + an numbers. If b = then n the sequence {n} is also convergent and lim b= lim an. n-x n-x arctan n III. lim 0. n-x 4n IV. Lemma: For any sequence {an} of real numbers, lim a2k = lim a2k+1 = L if and only if k→∞ k→∞ lim an = L; (LER). 14x By the lemma above, one can obtain that 1 2 (−1)n-1n if an + n n then lim an = 14x (a) I, III (b) III, IV (c) II, IV (d) - I, II, III (e) II, III, IV SIN n + 48 315 112 n n

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

8. Which of the following statements are true?
Vn? sin(n²)
1.
I. lim
n +1
n-00
II. Let {an} be a convergent sequence of real
a1 + ... + an
numbers.
If bn
then
n
the sequence {bn} is also convergent and
lim b, = lim ɑn.
n-00
n00
III. lim
arctan n
: 0.
n-00
4n
IV. Lemma: For any sequence {an} of real numbers,
lim a2k
k00
= lim a2k+1 = L _if and only if
lim an = L; (L E R).
n-00
By the lemma above, one can obtain that
1
jf an
(-1)"-1n
2
3
+
n
4
1
then lim |an|
n00
(a)
I, III
(b)
III, IV
(c)
II, IV
(d)
I, II, III
(e)
II, III, IV

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D. Suppose real sequences (sn) and (tn) are bounded. (That is, that their ranges are bounded sets.) i. Show the sequence given by (sn + tn) is bounded. ii. For any real number α, show that the sequence (α⋅sn) is bounded.
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