Let {a;} be a sequence in R. Prove lim an = 0, if and only if, lim Jan| = 0. n-00 If lim Jan] = 1, is it true that lim an = 1? Prove your answer. n-00 n-00

Q: Which of the following is the limit 00 of the sequence {a,},where an = 3? 3 8

A: The objective of the question is choose the correct option.

Q: Let {an}-1 be a sequence. Suppose lim |an] = |L| for some number L. Prove lim an is true. If it is…

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Q: 1 25. az = 1, an+1 =(a, +4) a) Since the sequence is recursive find an b) Find lim a,n n- DO c) If…

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Q: Let (an) be a sequence and suppose that lim a2n = L = lim @2n+1. n→∞0 n→∞0 Prove that lim, an ² =

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Q: . Let 7 = {rn}~ be a sequence of real numbers. Prove that the limitpoint set of T is a closed set.

A: Given that x¯ = xnn=1∞ is a sequence of real numbers . We have to prove that the limit point set of…

Q: 19. an = IN 2 τη

A: The sequence is given as an=12-n

Q: n=1 m sup,+∞ an = (lim infn an)+2. %3D n-00

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Q: A sequence {an} is defined by a1 = 2 and an+1 =3(1+an)/5+anfor all n ∈ N. Find the limit of such a…

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Q: 2n + 3 Which of the following is true about the sequence (a,) where a, = 5n + 4 (a) lim a, = +oo (b)…

A: Given that; an=2n+35n+4

Q: 10. Let {xn} be a sequence of real numbers. What does the limsup of this sequence mean?

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Q: Find a sequence of real numbers {a;} such that lim a, does not n-00 xist, but it has 3 subsequential…

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Q: Let (an) be the sequence defined recursively by a₁ = √3, an Prove that (an) is bounded. = √2an-1 +3,…

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Q: Suppose {Pi} and {qi} are sequences of real numbers such that lim pn = p and lim qn = q. Give an e-N…

A: suppose pi and qi are sequences of real numbers such that   limn→∞pn=p  and limn→∞qn=qSo by…

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.