Let (an) be a sequence such that (a)a, where a # 0. Show that la - a (3/4)a? lan-als and deduce that (an) → a. a) Does the result hold when a = 0? b) Does the result hold if the cube is replaced by the square?

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Q: A. Is the following sequence In n bounded above? an 2n B. If yes, what is it bounded by? If not,…

A: Here, the given sequence is {an} with an=ln n2n 1) Since ln n<2n the sequence is bounded above.

Q: 'n+1' lim en ) = e n+1 is the exponent of e) n

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Q: Show directly from definition that if a, is a positive real sequence with lim a, = 0 and lim f(r) =…

A: Solve.

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A: Given a1 = 5  and  an+1 = 5an

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Q: a Let (n) be a constant sequence in R (i.e. there is some constant c such that In = c for all n).…

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Q: Prove that there is no hooked Skolem sequence of order n= 8.

A: To Prove - Prove that there is no hooked Skolem sequence of order n = 8.         Theorem used - A…

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Q: Let (an) be a numerical sequence such that lim |an|¹/n = 7-8 ∞ gence of the series 5a," is (A) 83…

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Q: If ( a n) is a sequence that diverges to ∞, prove that Iim n --> ∞ (1/a n) = 0

A: Assume that the sequence an diverges to ∞. That implies, limn→∞an=∞.

Q: Assume that the sequence converges and find its limit. a 1 = 8, a n + 1= 8an O 1 64 8 -8

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Q: Use iteration to guess an explicit formula for the sequence: ek = 5ek-1 + 2, for all integers k ≥ 2,…

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Q: 2. Give an example of an unbounded sequence {an} of negative terms such that an does not diverge to…

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Q: determine whether the sequence is monotonicand whether it is bounded. an =(2n + 3)!/(n + 1)!

A: To find the sequence is monotonic or bounded. Consider

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Q: Let a, be recursively defined by ao = 15, an+1 + for an n 2 0. Show that the sequence converges, and…

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Q: Use iteration to guess an explicit formula for the sequence defined by an = an-1 + n for all…

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Q: Use the iteration method to write an explicit formula for the sequence that satisfies ao = 5 and ar…

A: The iteration method is the method in which the formula for a sequence can be obtained by…

Q: Consider the sequence defined by an a, = * for every integer n 2 0. Find an alternative explicit…

A: Floor and ceiling function: Definition of the Floor: We define x = the biggest integer less than or…

Q: Find a formula for the general term an of the sequence 2 5 8 11 14 3'9 27' 81' - 243' assuming that…

A: find Your Solution Below

Q: Assume that the sequence converges and find its limit. 72 a, = 4, an +1- 1+ an for n = 1, 2, 3, . O…

A: Given query is to find the limit of the sequence.

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Q: Let {an}1 be a sequence such that there exist numbers a and N such that, for n > N, an = a. Prove…

A: We know that the sequence an converges to α when for any ε>0 there exists K such that for n>K,…

Q: Given: Cn . Is the sequence {c, , convergent or divergent? tan-in

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Q: Find a formula for the nth term of the recursively defined sequence where a, =1, an+1 an for n21. 4"

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Q: Let a, be recursively defined by ao = 15, ans1 = ;(a, + 2) for n20. Show that the sequence…

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Q: Let an = n 2 1 In(2" + 1) Determine if the sequence {an}n=1 is convergent or divergent 00

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Q: Let an be a sequence defined by a1 = 0 and a2 = 1 and for n > 3 an = (n – 1)(an-1+an-2). %3D

A: Here, the terms given in the series are, a1=0,a2=1an=n-1an-1+an-2 Calculate the third, forth and…

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Q: Let (x,) be a convergent sequence of real numbers and let (a,) and (b,) be subsequences of (x,) with…

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Q: determine the limit of the sequence or show that the sequence diverges.

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Q: Determine whether the following sequence is convergent or divergent. Justify your answer. {1+u^ =…

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Q: 1. Consider the sequence {an}, where a, e [a, b] for all n e N. Show that is convergent.

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Q: (b) Define another recursive sequence (hn) by removing fi, f3, f5,... from the bonacci sequence.…

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Q: Explicitly calculate the first six val- ues of the factorial sequence using the formula ao = 1 ап —…

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Q: Find an explicit formula for a sequence of the form  a1,  a2,  a3,     with the initial terms given…

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Q: Find an explicit formula for a sequence of the form a,, a,, az, with the initial terms given below.…

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Q: Calculate the lim sup and lim inf (if they exist) for the sequence (a,) given by: a, =7+(-1)".

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Q: Find a closed formula for the sequence with recursive : 2 = 1 and a2 definition an 2an An – 2 with…

A: Given  an=2an-1-an-2  , a1=1 , a2=2 we have to find the closed formula for this sequence.

Q: Compute the first five terms of the sequence define by the formula T, = 2n2 – 5n + 1 For what value…

A: Given Tn  =  2n2 - 5n +1 T1 = 2×12 - 5×1 + 1 = -2 T2 = 2×22 - 5×2 + 1 = -1 T3= 2×32 - 5×3 + 1 = 4…

Q: dentify the sequence as geometric ór arithmetič. (r, = 11000 Im = 0.7rn-1

A: We can check the series as below by definition of geometric series

Q: and (b) use a graphing utility to graph the first 10 terms of the to show that the sequence with the…

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Q: 4. Define a sequence by an+1 = -an?, with -1 < ao < 0. Prove that {an} is increasing and bounded…

A: Consider the terms of the sequence put n=0 a1=-a02 here -1≤a0&lt;0 So -1≤a1&lt;0 but certainly a1≥a0…

Q: Give a recursive definition of the sequence {an}, n = 1, 2,... if: a. an=4n-2 b. an=1+ (-1)"

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Q: 1. Prove that the sequence diverges but it has a convergent subsequence: (-1)" 2+(-1)", (a) (b) X =…

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Q: a) If some subsequence of a monotonly decreasing sequence {an} is convergent to 5, then {an} is also…

A: Theorem: A monotone sequence of real numbers {an} having a convergent subsequence with limit a, is…