Consider the following sequences defined for nEN, n21 9 +28 {an} 22" {bn} = { In(22 + 2) – In(9 + 2)} It can be stated with certainty that: a) {an} and {bn} converge b){an}converge and {bn}diverge c) {an} and {bn}diverge d){a,}diverge and {bn}converge

Q: Prove that if the sequence {an} is divergent and the sequence {bn} is convergent, then the sequence…

A: Given: The objective is to prove that if the sequence {an} is divergent and the seqence {bn} is…

Q: Does the series 1+ 1/8+1/27+1/64+1/125+...(reciprocals of the perfect cubes) converge or diverge?…

A: we have to test whether given series is convergent or divergent

Q: Prove that following sequences converges to zero. 2n Xn n² + Tt

A:

Q: Suppose a Cauchy sequence {xn} is such that for every M ∈ N, there exists a k ≥ M and an n ≥ M such…

A: every cauchy sequence of real numbers is convergent. And a sequence can have at most one limit.

Q: Assume {Pn}, and {rn} are Cauchy sequences in R. Using the definition of a Cauchy sequence prove…

A:

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…

Q: Prove that if a sequence (an)1 is not bounded, then there is a subsequence (akn)-1 which either…

A:

Q: Consider the series E, Ck where 0 < Ck < ; for each k E N. If it is given that Ek=1 Ck _is…

A:

Q: Use a Maclaurin series to compute the limit.

A:

Q: En3D1 In(n) 2 +1

A:

Q: Define the sequence {Sn} by Sn= [ 1 / (2)(1) ] + [ 1 / (3)(2) ] +   .............  + [ 1 / (n+1)(n)…

A: The nth term of the partial sum is given by

Q: If a, converges and a, > 0 for every n, show that 4converges. What can you say about a for any…

A: Given: ∑n=1∞an converges and an&gt;0 To prove: ∑n=1∞an2 converges. Proof: ∑n=1∞an2=∑n=1∞an2 Since…

Q: Question * Every Cauchy sequence in the Euclidean metric space R" with n a positive integer is…

A: True. I have proved using definition of cauchy and convergent sequence.

Q: 1.Determine the convergence of the following sequences {a,} = {In (#)}

A:

Q: Show that the series -1 (2n+1)" converges absolutely. in=1

A: We will check its convergence by ratio test .

Q: Suppose Sanx",Σ bnan converges for all x in (-1,1). Σana" = Σbnan in a subset E of (-1,1) implies…

A: It is given that,  ∑anxn and ∑bnxn are convergent for all x in (-1, 1). Then,  ∑anxn = ∑bnxn in a…

Q: Determine the convergence/divergence of the following sequences/series. 1. {an} = {1- ( In n | In…

A: We use the techniques of analysis to solve this problem.

Q: Let (an) be a sequence in which every term is from some given finite set R. Prove that (an) has a…

A:

Q: Prove that the sequence {n} defined by Xn = 1+ is a Cauchy sequence.. ¹ + 1 + 1 + 4 9 +

A: Introduction: A sequence is called a Cauchy sequence when its terms eventually become arbitrarily…

Q: {In (n+10) – In (n+5)}°

A: an =ln(n+10)-ln(n+5)an =lnn+10n+5limn→∞ an =LIf the limit exists and finite, then the sequence is…

Q: If an < bn and if En=1 an converges, then 1 bn also converges. True False

A: we can solve it by using comparison test.

Q: The series Zan +... converges to 1 16 8 n = 1 Select one: True False

A: given ∑n=1∞ an = 12 -14+ 18-116+...   12-14+18-116+... ∑n=1∞-1n+12n= ∑n=1∞- -12n

Q: If an and bn are sequences such that 0 < an < bn and bn converges, then an converges. Select one: O…

A: We will solve this question using a example.

Q: @ suppose {an} and { bn} are sequences such that {an} and { ant bn'} converge.prove that {bn}…

A: To prove (a)  Given that an converge and an+bn converge. The objective is to show that bn converge.…

Q: 8. Suppose that for sequence of real numbers {an} and {b,} that O< bn+1<an < bn, Vn EN Prove that…

A:

Q: Let (pn)neN be a strictly increasing sequence of positive integers and x be a positive real number.…

A: As per the question we are given a strictly increasing sequence (pn)n ∈ N and a positive real number…

Q: (a) Consider the definition of Fibonacci series: where Fo= 0, F₁ = 1, F₂ = 1, and the recursive term…

A: Given the Fibonacci series as Fn+1=Fn+Fn-1 Also given F0=0F1=1F2=1

Q: n! 1 · 3 · 5 ··· (2n − 1). +∞ n=1

A:

Q: Give an example of two sequences (an) and (bn) such that all of the following are simultaneously…

A: To Give: An example of two sequences an and bn such that all of the following are simultaneously…

Q: ) Pick your favourite real power series o a,r". Show that for this series, the equivalence 00 Σ anx"…

A: c) Take an=1n!. Then the series ∑n=0∞anxn=∑n=0∞xnn! and ∑n=1∞nanxn-1=∑n=1∞xn-1n-1!. To show that it…

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

A:

Q: Test the llawing Seri es for being diverge Converge そ ) n +2 (-1) 30-1 (en)!

A:

Q: Use Theorem 9.11 to determine the convergence or divergence of the p-series. 1 + 5 16 1 + 5 81 1 + 5…

A: 1+1165+1815+12565+16255+... 1+1245+1345+1445+1545+... 1145+1245+1345+1445+1545+...…

Q: 1. Determine the convergence of the following sequences. C. 2n+5 2n 3n

A: I have given the answer in the next step. Hope you understand that

Q: Determine whether the sequence converges or diverges an = In(n + 1) – In(n) %3D lim an %3D n- 00

A: Given sequence is an=lnn+1−lnn We have to find the limn→∞an Use property lnm−lnn=lnmn…

Q: Show that the fractional parts of a given sequence {annz1 is uniformly distributed if and only if 1…

A:

Q: Suppose that the sequence {an} is monotone. Prove that {an} converges if and only if {a%} converges.

A:

Q: Let an n > 3. nvInn (b) Without computing any limits, use the Monotone Convergence Theorem to…

A:

Q: 00 n³n² COS 5n³ +3, 1

A: See the answer below

Q: Consider the series E, ak where ar > 0 for each k E N. If it is given that c=1 Ek=1 ak is…

A:

Q: If (x,) is a sequence in R such that the series 2(x is convergent, then the series 2 (x) n=1 must be…

A:

Q: Consider the definition of Fibonacci series: where Fo = 0, F1 = 1, F2 = 1, and the recursive term is…

A: Given: Consider the Fibonacci series F0=0, F1=1, F2=1 and the recursive term is defined as…

Q: Prove if ( a n ) is a Cauchy Sequence and ( a n ) has a subsequence that converges to L, prove that…

A:

Q: What about the sequence an =16 + (10n2+1)/en for all n.  Is it monotone? Does it converge?  If so,…

A: Given a sequence witn nth term is shown belowA function is said to be monotonic if it is entirely…

Q: Let {X,} be an independent Bernoulli sequence with parameter p, and let np, A < o as n oo. Verify…

A: Verifying that the Lindeberg condition fails for Xn

Q: Let s, = a1 + az+ ·…+a, be the nth partial sum of a series Σ am- The series converges if and only if…

A:

Q: 2. By taking the limit N 0o, argue that if -12" azn converges then so does -1 an. (Hint: apply the…

A:

Q: Use the Monotone Convergence Theorem to show that the following sequence 1.3.5(2n – 1)1 2 .4 . 6 ...…

A: Given: 1.3.5...(2n-1)2.4.6...(2n)n=1+∞