2. Use the definition to prove that if (xn) is a sequence of real numbers which converges to -3, then the sequence (2In + 1) converges to -5.

Q: Ex. Show the seguence whose nth term is In (n*) an %3D converges. 5n

A:

Q: Show that the sequence {cn} = {(−1)n 1/ n! } converges, and find its limit

A: The nth term of the given sequence is .

Q: Prove that the sequence {cn} converges to c if and only if the sequence {cn- c} converges to 0.

A: To prove that the relation between the limits of the two given sequences

Q: Prove that the sequence ((-1)")=1 does not converge to any number.

A: Solution :- The given sequence is                          {  an }   =   { (-1)n}  then , we have to…

Q: Prove that the sequence (xn) = (-1)"+1 [3 .does not converge to any real number

A:

Q: Prove that the sequence given by 2.4. 6.(2n) Un = 3.5.7.…(2n + 1) converges and find its limit.

A: Given query is to prove that sequence converges.

Q: 2. Show that if a > 0, then the sequence (²x²e¬n) converges uniformly on the interval [a, ∞), but…

A:

Q: II. Prove if X and Y are the sequence of real number and converges to x a real number then product…

A: Given that X and Y are the sequence of real number And X and Y converges to x a  real number. The…

Q: 1 Prove that if a sequence {a„} converges to a, then the sequence {[an]} converges to |a|. Find…

A: Suppose {an} converges to a. Prove that an converges to a. Also, check the converse is true.

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

A:

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

A:

Q: B. Prove that limx → c f(x) = o if and only if for every sequence aŋ that is in the domain of f and…

A: Consider a function fx. Suppose that limx→cf(x)=∞. That is for all N>0, there exists a δ>0,…

Q: Suppose that {a, } is a monotone sequence such that a, 10 or increases- in this case we can't say…

A: We need to choose correct options.

Q: Find all values of x such that converges. n! n=1

A: Given that, series ∑n =1∞  xn2n! Compare the series ∑n =1∞  xn2n! with its standard form ∑n =1∞  an…

Q: 4. Show that the sequence {fn}, where fn(x) = xER 1+n°x2 is point-wise convergent but is not…

A:

Q: Find all values of x such that ΣΗ converges. n! n=1

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Q: Suppose a,>0 for all n and 2 a, converges , which of the following must converge n = 1

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Q: Is {3n/(3n-1)} n=1 to infinity, a convergent sequence in R with the usual metric dR(x,y) = abs(x-y)?

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Q: . Find all values of x for which пх" n=1 (n + 1)(2x + 1)" converges absolutely.

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Q: Suppose that the sequence {an} converges to a, where 0 < a < 1. Prove that the sequence {a}…

A:

Q: 4. Show that the sequence {fn}, where nx fn(x) = xER 1+n°x is point-wise convergent but is not…

A:

Q: We say that a sequence of real numbers (xn)nɛN converges to the real number а, if Ve > 0,3N E N, Vn…

A: Given sequence is xn=-1n+2-n. Definition: A sequence of real numbers xnn∈N converges to the real…

Q: 14. What does it mean that a sequence {xn} tends to infinity?

A: What can we say if the sequence tends to infinity .

Q: Prove that the sequence (an) = (-1)"+' does not converge to any real number. %3D

A:

Q: Use only Definition 3.1.1 (or its negation) to prove the following: 2. Prove that the sequence…

A:

Q: 20

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Q: Prove using the ϵ−n0 definition that the sequence Xn=(9−7n)/(8−13n) converges, and find its limit.

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Q: 1. Recall the definition of the limit of a sequence of real numbers (xn) = (x1, X2, X3, . . .) "(Tn)…

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Q: Suppose (xn) is Cauchy and (yn) is bounded and monotone. Can we say anything about whether the…

A:

Q: 3] Prove that the sequence (x,) = (-1)"+' does not converge to any real number.

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Q: b) E, 2n-1

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Q: The sequence (tanh n) converges to2

A:

Q: 2. Use the definition to prove that if (xn) is a sequence of real numbers which converges to -3,…

A:

Q: Assume a sequence xn satisfies 0 1 for any n e N. Prove that 1. xn is an increasing sequence. 1 2.…

A: The given sequence is: 4xn+1(1-xn)≥1, where0&lt;xn&lt;1 for any n∈ℕ To prove: 1) xn as an increasing…

Q: Suppose (xn) is Cauchy and (yn) is bounded and monotone. Can we say anything about whether the…

A: Convergent:- If limn→∞an=l, then we say that the sequence an is converges to l. Given the sequence…

Q: Prove that or 「+ X。- ズー 2. 71 number J2 Converged to

A:

Q: 2. a) Show that for any positive integer k, the sequence defined by pn = converges linearly to p =…

A: Since you have asked multiple questions, we will solve first question for you. If you want other…

Q: true or false , prove your answer Suppose the interval (3, 3.99999) contains infinitely many terms…

A: Theorem:If a sequence is monotonic increasing in an interval then it's converge to it's supremum.

Q: 2- Suppose that (xn) is a sequence such that xn > 0 for all NE N. 1 lim n 0 Xn Show that; Xn → 0 as…

A: I have proved using Apsilon delta definition

Q: Find a sequence X, such that it does not converge to any real number, but Y, does converge to some…

A: Consider the sequence Xn=(-1)n. Then Xn={-1,1,-1,1,-1,1,-1,1,...}. So, Xn does not converges to any…

Q: For any n N, define the function Xn RR by : if x ≥n Xn(2)= 0, if a <n. Show that the sequence of…

A: We will use definition of limit of sequence of function.

Q: 2n+1 4. Use the definition to prove that ) converges.

A:

Q: 4. Show that the sequence { fn}, where nx fa(x) = xER 1+n°x2 is point-wise convergent but is not…

A:

Q: an Show that if lim bn =0 and =1 n- 00 converges then En=1ªn converges too. n=D1

A: We have to solve given problem:

Q: Let (x,) be a sequence of real numbers. If for some x e R, (x,) converges to both 7x + 2 and 2x - 13…

A: Which of the following is correct?

Q: Show that 2"x" converges if |x| <. n=1

A:

Q: an = 0 and bn Σ-1 bp Show that if lim n converges then En-1an converges too.

A: Given that:            limn→∞anbn=0 and ∑n=1∞bn converges. Prove ∑n=1∞an converges.

Q: Let (X,) be a sequence of real numbers. If for some x e R, (X,) converges to poth 7x + 2 and 2x - 13…

A: which of the following statement is true.

Q: Prove that a sequence of positive real numbers {r„}1 tends to 0 if and only if the sequence {},…

A: We have to prove a sequence of positive real numbers xnn=1∞ tends to 0 ⇔ 1xnn=1∞ tends to ∞.

Q: Prove that the sequence {sin(nn/3)} does not converge.

A:

Q: The sequence an = n- -In+1vn +3 converges to the limit %3D A) -3 B) -1 C) 1 D) +2 E) -2

A:

Q: Show that the infinite sequence 5. an = √ (n2 + 1) − √ (n2 − 1) (n ≥ 1) converges by showing that it…

A: Given sequence is  an = √ (n2 + 1) − √ (n2 − 1) (n ≥ 1)

Q: 5. Show that the sequence Pn = 1 converges linearly to p 0, and the sequence Pn = 10-2" converges…

A: