If {a} be a sequence such that lim an +1 an = 1, where | 1|< 1, then lim an = 0.

Q: Give an example of a sequence an whose set of subsequential limits S = {a €RU{-0}U{x} : a = lim ak,…

A:

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…

A:

Q: If an Is an increasing sequence and every term a, is less than 10,

A: Given:- If an is an increasing sequence and every term an is less than 10, then lim      an=10n→∞

Q: Determine whether the sequence {a,} converges or diverges. an = (-1)"_n + 3 4n + 1 converges O…

A:

Q: Show that the following sequence is increasing:

A:

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

A:

Q: (6) Let {an} be the sequence defined by an+1 = √6an 2, a₁ = 1. Show that {an} is converger and find…

A:

Q: If the sequence {a}, is convergent, then the sequence {x} defined by a₁+as+as++an-stan is also…

A:

Q: 1 2 + xn (a) Prove by induction that în > 0 for all n € N. (b) Prove that (n) is a contractive…

A: (a) We have to prove by induction that xn &gt; 0 for all nЄ N.(b) We have to prove that (xn) is a…

Q: If lim f(x) =L, then lim an = L. n-00

A: Given: Let an be a sequence and f be the function such that f(n)=an for n≥1 Also,  limx→∞f(x)=L To…

Q: Exercise 4 We say that the sequence {a} is unbounded from above if there exists a subsequence (an,}…

A:

Q: Let {an} be a real number sequence. Prove that                             {an}= {1+(1/2n)}…

A:

Q: For the sequence {a ,} if a,→ L, and f(x) is a function defined at all a, then f(a ,)→ f(L) Select…

A:

Q: 2) Show that the sequence defined recursively by 3 4 An aj = 5 аn+1 satisfies 2 < an < 5 and is…

A:

Q: Let p be a given Natural Number. Give a sequence (XN) that is not a Cauchy sequence, but it…

A:

Q: (a,) be a Cauchy sequence of nonzero real numbers. Which of the following statements about (a,) may…

A:

Q: If a sequence an is strictly increasing and not bounded above, then the limit of sequence does not…

A: To state whether true or false.

Q: Let the sequence 9n be recursively defined by go = 0, 91 = 1, 9n = 9n-1 +29n-2 for n ≥ 2. Show that…

A: Given: The recurrence relation, gn=gn-1+2gn-2, g0=0, g1=1 for n≥2 To show: limn→∞gn=∞

Q: 2. Show that if a sequence {an} satisfies lim a2n lim a2n+1 = s then lim an = s. n 00

A:

Q: Give an example of a divergent sequence {a,} such that lim Ja,| converges. n 00

A:

Q: Let {an} be a monotonic sequence of real numbers. If = a, and an+1 = a + which of the following is…

A: In the given question we have to find the limit of the sequence where is defined as…

Q: If {b,}-1 is a bounded sequence and lim,→∞ An = lim,-+∞ anbn? Justify. 0, what can you say about

A: Given that  bn is bounded sequence, so, by definition of bounded sequence  ∃B&gt;0 such that bn&lt;B…

Q: 1. Consider the sequence {an} = {-250, 100, -40, 16, 32). (a) Determine whether the sequence {a}…

A:

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…

A:

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: lim 3n+2 2n + 1 = 3/2

A:

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

A:

Q: Let {xn} be a bounded sequence of real numbers. Prove that {xn} converges if and only if lim inf xn…

A:

Q: The first term of a converging sequence {an} is given by, a1 = 0 and the remaining terms can be…

A:

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

A:

Q: Give the definition of a subsequence of a sequence Xn.

A: the definition of a subsequence of a sequence Xn.

Q: 1 3 Consider the sequence {an} where a1 = 2 and an+1 = an + 2 Show that the sequence is monotonic.…

A:

Q: Please give an cxample of a sequence (a,) such that: (a) (an) is not bounded above. (b) (an) is not…

A:

Q: Please give an cxample of a sequence (a,) such that: (a) (an) is not bounded above. (b) (an) is not…

A:

Q: If lim an # 0, then the sequence {a,} is divergent. n+0 O True False

A: