Suppose that the sequence {n} is such that xn 20 for all n eN and that lim(-1)"xn exists. Show that {xn} converges.

Q: Let (Sn) be convergent sequence, and suppose lim s, > a. Prove there exists a number N such that n>…

A: Given sn is a convergent sequence, and suppose limn→∞sn>a. Since sn is a convergent sequence,…

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: Suppose that the sn satisfies both limn→∞s2n = -1 and limn→∞s2n+1 = -1. (That is, the sequence given…

A:

Q: Determine whether the sequence converges or diverges. If it converges, find the limit. 2. An In(n+1)…

A: 2.an=lnn+1-ln n

Q: Show that the sequence defined by 1 a₁ = 1 an is increasing and an < 3 for all n. Deduce that {an}…

A:

Q: Determine whether the sequence is convergent or diver- gent. If it is convergent, find its limit.…

A: We have ∑n=0∞an=∑n=0∞9n+110n=∑n=0∞9·9n10n=9∑n=0∞9n10n=9∑n=0∞910n=9·Sn

Q: Consider the sequences (an) and (bn) given by an= (−1)nn2, and bn= 2n. Decide if the sequence…

A: Convergency for an:we can find limitwe know that (-1)^n is oscillating This value can be either -1…

Q: Determine whether the sequence is convergent or diver- gent. If it is convergent, find its limit. 2…

A: Given :  The sequence an = 2 + n31 + 2n3   To determine the sequence is convergent or divergent.

Q: Determine whether the sequence {/IIn + 15 – V11n} converges or diverges. If it converges, find the…

A:

Q: Determine whether the sequence converges or diverges. If it converges, find the limit. an = ln(6n2 +…

A: Use logarithmic rules as follows.

Q: Let {an} be a monotonic sequence such that an ≤ 1. Discuss the convergence of {an}. When {an}…

A:

Q: iii tan 3n + 1 n=1

A:

Q: Determine the convergence or divergence of the sequence with the given nth term. If the sequence…

A: To check whether the sequence is convergent or not, first evaluate the lim an. If the limit value is…

Q: 10. Determine if the following sequence an = In(n)/In(2n) converges or diverges as limit no. Find…

A:

Q: 3. Determine if the sequence xn = 47² is convergent or not and find its limit if it exists n²+1

A: We want to find the given sequence is convergent or divergent.

Q: Suppose that the sequence {an} is monotone and that it has a convergent subsequence. Show that {an}…

A:

Q: Suppose that a, 2 0 and b, 2 0 for all n> N where N is an integer. If an lim = 00, what can you…

A:

Q: 5* Determine whether the sequence defined by a* %3D 8* +1 converges or diverges. If it converges,…

A:

Q: Determine whether the sequence converges or diverges. If it converges, find the limit. (2n 1)! (2n +…

A: To check whether the given sequence is convergent or not , we first find the limit of the given…

Q: Please help

A: The given recursively defined sequence is

Q: Show that the series n" n' is convergent. Deduce that lim (2n)! (2n)! n=1

A:

Q: Suppose that the sn satisfies both limn→∞ s2n = 2 and limn→∞ s2n+1 = 2. (That is, the sequence given…

A: Step:-1 Limit of a sequence:- Let f(n) be a sequence of real numbers and let l be a real number then…

Q: 7. Determine whether the sequence, n 1+4n² 1+n² > converges or diverges. If it converges, find its…

A:

Q: In(8n2 + 8) – In(n²

A: We have to test the convergence

Q: Show that 1-Every Convergent Sequence is bounded sequence. 2- Show that the converse need not be…

A: As per our guidelines, we are supposed to solve first two multipart question. For other question…

Q: Consider the iteratively defined sequence (x,), where x41 = 3 - 2 and x = 3. Use the Monotone…

A: Monotone convergence theorem: If a sequence monotone and bounded then it is convergent.   If the…

Q: Let (Xn)n and (Yn)n be two convergent sequences with limits x∗ and y∗. Prove that (a) xn + yn is a…

A:

Q: Determine the convergence or divergence of the sequence with the given nth term. If the sequence…

A: Given,  an=(4n)!(4n-1)!

Q: converges for all * > 0 because lim a - In n 0, for all * > 0- n=2 True False

A: Need to check given statement is true or false.

Q: Show that the series 71 n' (2n)! is convergent. Deduce that lim 0% (2n)! n=1

A: We have to show that the series ∑n=1∞nn2n! is convergent. Let an=nn2n! Now,…

Q: Let ₁1 and In+1 = limit. 1 for n N. Show that this sequence is convergent and find its In + 2

A:

Q: Suppose the sequence {a,} is convergent, in particular lim,-o an = L. Does this mean that every…

A:

Q: determine whether the sequence converges or diverges. if it converges finds the limit.…

A:

Q: Determine whether the sequence is convergent or divergent. If it converges, find the limit. {n² tan…

A: Here we have to check the convergence of the sequence.

Q: The sequence (tanh n) converges to2

A:

Q: 10. Determine if the following sequence a, = In(n)/In(2n) converges or diverges as limit no. Find…

A:

Q: Show that the sequence ffn}, where fnox) = ne is uniformly convergent nx+1 on [a, b], aso but is…

A: Given: fnx=nxnx+1 To show that the given sequence is uniformly convergent on a,b, a&gt;0

Q: Suppose {xn} is a bounded sequence. Define bn where lim inf xn :_(n→∞) = lim inf bn :_(n→∞). Show…

A: It is given that xn is the bounded sequence. bn:=infxk: k⩾n Wherelimn→∞infxn:=limn→∞bn To prove bn…

Q: Which of the sequences whose nth terms appear converge, and which diverge? Find the limit of each…

A: an=ln2n+1n Now, limn→∞an=limn→∞ln2n+1n=limn→∞12n+1·21       By using L'Hospital's…

Q: If possible, give an example of 2 convergent sequences an and bn such that E(an + bn) diverges.

A:

Q: 1.) Determine whether the sequence converges and if it does then find its limit: (п-1)! an = n tan…

A:

Q: Determine whether the sequence converges or diverges. If it converges, find the limit. an-In(n 2)…

A: The nth term of the given sequence is,

Q: Determine whether the sequence converges or diverges. If it converges, find the limit.  an=3n/2…

A:

Q: 14. Suppose (an) is a bounded sequence such that all of its converging subsequences converge to the…

A:

Q: 7) Determine whether the sequence (1 - )} 3 4n converges or diverges. If it converges, find the…

A:

Q: Use Taylor series to evaluate the limit. lim x -4 -1 O 4

A: Given: limx→∞x2e4x2-1 ....(1) From Taylor's series we have e4x2=1+4x2+124x22+164x23+...…

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: an = In(3n2 + 1) – In (n² + 1)

A: For the solution follow the next step.

Q: Show that the series n" (2n)! is convergent. Deduce that lim 0. (2n)! n=1

A: