Let (an) be the sequence with an = (-1)” +212/2. (i) Use the fact that 1/n² → 0 as n → ∞ and Proposition 10.2.2 to show that 1/(2n)² → 0 and 1/(2n − 1)² → 0 as n →∞. (ii) Show that 1 and -1 are limit points of the sequence (an). (iii) Use Proposition 10.2.2 to show that the sequence (an) does not con- verge.

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: Consider the following sequence where f(1)=23​. 23,32,41,50,59,…​a) What is the common difference,…

A: The given sequence

Q: 1. Find the limit of the sequence {√3, √√3+√√3, √√3+ √√3 + √3,...}.

A:

Q: (n + 1)! Determine the limit of the sequence a, n! 3 1 2 1 1

A: find limit of sequence

Q: a) Let (an) be a sequence of real numbers and a ER. Suppose an → a. Show that a1 + a₂ + n does. ...…

A: (a) Given that <an> is a sequence such that <an>→a, To Prove:  a1+a2+...+ann→a as n→∞.…

Q: Find the limit of the sequence as n approaches infinity: In 1 an 2n + 2

A:

Q: 12. Suppose the sequence ao, aj, a2, a3, ... is defined as follows: ao = 0, aj = 1, and for n > 2,…

A:

Q: Define a sequence of ordered pairs (x;, y;) as follows: (xo, Yo) = (3,1) %3D Xn+1 = 3xn + 4yn, n > 0…

A:

Q: A sequence is defined by f(1) = 3 and f(n) = 2. f(n –1) for n > 1. Which of the following statements…

A: Here given  f1=3,fn=2·fn-1

Q: Find a function  f(n)  that identifies the nth term  an  of the recursively defined sequence, as  an…

A:

Q: 3. Prove that the sequence with nth un (a) is monotonic increasing, (b) is bounded above, (c) is…

A:

Q: Suppose we have a recursive sequence ƒ1, ƒ2, ƒ3, . . .. For the purposes of this problem, it does…

A: Observe that in the inductive step, to determine fk+1, you need to know two step back fk-1. So, we…

Q: (2) Let an be the sequence defined by: a₁ = 1 a2 = 2 an an-1 + 2an-2 for n ≥ 3. (a) Find a3,.. (b)…

A: Given that a1=1 , a2=2 and an=an-1+2an-2  for n≥3

Q: Consider the arithmetic sequence S, where the i-th term of S is defined as follows for positive…

A: Solution; The arithmetic sequence is defined by Si=2+6i-1 We want to prove by induction that the sum…