b) With justification give an example of a sequence such that - 1, 1 are its two limit points and no member of the sequence belongs to the interval |- 1, 1).

Q: (-2)" Determine the limit of the sequence a, = -2 1 - 1 Divergent O O O O O O O

A:

Q: Give an example of a bounded sequence {a„}_1 having exactly 2 limitpoints.

A: we have to find a sequence such that it has two limit points.

Q: Let x, = 3. Using the algebra of limits and standard results on limits to establish/justify if the…

A: Consider a sequence an. Then if the limit limn→∞an exist then the sequence converges otherwise the…

Q: Use any method to prove that if X = (xn) is a Cauchy sequence of real numbers, then () is also a…

A:

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: 10. Use the definition of the limit of a sequence to prove that (3n+1 lim 2n + 5/ for all positive…

A: we have to use definition of limit of sequence.

Q: . Suppose real sequences (S,) and (t,) are bounded. (That is, that their ranges are bounded sets.)…

A: Given the real sequences sn and tn are bounded. We have to show that The sequence given by sn + tn…

Q: Assume that the recursively defined sequence converges and find its limit. 15 a, = 1, an + 1-2+ an…

A:

Q: Give an example of a bounded sequence without a limit.

A: If any sequence has a limit, then that sequence is said to be convergent. So here the objective is…

Q: 22: Establish the divergence of the sequence x, =(-1)"

A:

Q: n2 – 1 - at the sequence ɑn is eventually stric n2 + 2 so show that {a,} is bounded above. What can…

A:

Q: 3.) Determine whether the sequence is monotonic and whether it is bounded. sin /n

A:

Q: Assume that the recursively defined sequence converges and find its limit. a1 = - 2, an +1 = /14 +…

A:

Q: Consider the sequence {. n+2 1. Determine the number of terms that are not within ɛ = 0.0002 of 3.…

A:

Q: a) Show that the sequence a, = (-1)" +- does not converge. b) Let b, be a sequence such that for all…

A: Given the sequence an = ( -1 )n + 1/n . It has two subsequences a2n = 1 + 1/n converging to 1 and…

Q: Let {bn} be a sequence whose limit is 3. Show that if B is an upper bound of this sequence, then 3 <…

A: We will use the definition of limit.

Q: Among these statements, which are true and which are false There is a sequence with an infinite…

A: 1. TRUE x(n)=1-((-1)^n) which has Range ={0,2} and for every alternate terms we have infinite zeroes…

Q: Exercise 4. Use the definition of limit to prove that the sequence {n – - is divergent. -

A: To prove: the sequence n-1nn=1∞ is divergent. limn→∞n-1n=limn→∞n-limn→∞1n…

Q: Prove that the sequence is increasing.

A: In this question, we have given a sequence  an=(1+1n)n+11+1n To prove the sequence is increasing by…

Q: C) Prove that a Cauchy sequence of real numbers is bounded.

A:

Q: c) Prove that a sequence in R can have at most one limit that is “uniqueness of limits"

A: consider  Xn be a sequence in ℝ

Q: Give an example of a bounded sequence that has a limit.

A: Give an example of a bounded sequence that has a limit.

Q: 3. Evaluate the limit of the sequence or state that it does not 2" + 5**1 exist. 5"

A: Limits exists and limit value is 5 .

Q: Give an example of a nonincreasing sequence with a limit.

A: The nonincreasing sequence is a sequence {An} of the real numbers that never increases that is An+1…

Q: consider the sequence  1, 1/2 ,1, 1/3 ,1 ,1/4, 1,.........  determine the limit superior and limit…

A: Given sequence 1, 1/2 , 1 , 1/3.. The limit superior is 1. As every number of sequence &lt; 1+k for…

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

A:

Q: Every convergent sequence of real numbers is not bounded ylgn İhi

A: Every convergent sequence of real numbers is not bounded.

Q: 6. Show that the sequence (5)" is Cauchy by using the Cauchy definition.

A:

Q: Prove that sequence {an}= {(-1)n} does not have a limit.

A: According to the given information it is required to show that {an}= {(-1)n} has no limit.

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

A:

Q: 3. Prove that if (a, } and (b,} are Cauchy sequences of rational numbers satisfying lim (a, - b, = 0…

A: 3.

Q: use Theorem 1 to determine the limit of the sequence or state that the sequence diverges.an = 12

A: To use theorem 1 to determine the limit of the sequence.

Q: Prove or disprove that if a sequence {x,} satisfy x,+1 – xn → 0 then it is Cauchy

A: We have to solve given problem:

Q: Consider the sequence  1,1/2,1,1/3,1,1/4,1, ... Determine the limit superior and limit inferior of…

A: Given sequence is  1,12,1,13,1,14,1,....... By observation , it can be seen clearly that 1 repeats…

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

A:

Q: Show that the set of limit-points of a sequence is a closed set.

A:

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

A:

Q: Evaluate the limit of sequence ak  = k-1/k-2

A: We have to find limit

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

A:

Q: Evaluate the limit of the sequence or state that it does not exist. an = (π)/(2) - tan-1n

A: To evaluate the limit of the sequence an=π2-tan-1n

Q: Is it true that a sequence {an} of positive numbers must converge if it is bounded from above? Give…

A:

Q: 7,11 Write out the first five terms of the sequence, determine whether the sequence converges, and…

A:

Q: Use the definition of the limit of a sequence to prove that 3n +1 lim 2n + 5 3 %3D 21

A: We need to use the definition of the limit of a sequence. we have the definition of limit as,…

Q: 4. Determine if the sequence is bounded and if it is monotonic. Determine if the sequence converges…

A: This question is related to sequence and series.

Q: Show that the real sequence ( 0,-1,2,-3 , .... , (-1)nn, .... ) where n is a value from the set of…

A: Consider the given sequence,

Q: Use the Theorem of Sequence Defined by a Function to determine the limit of the sequence or state…

A: Topic - limits and series

Q: Give a sequence of positive numbers which is NOT bounded and has exactly one limit point.

A: A real no.p is said to be a limit point if every neighborhood of p contains an infinite no. Of…

Q: Assume that the recursively defined sequence converges and find its limit. a, = - 17, an + 1 = /35 +…

A:

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 ∞.

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.