25. Let an := n 2 1. Which of these conditions must be met if a, is to converge n+1 to 1? I. for any e > 0, n > II. for any e > 0, | II. for any e > 0, | kevn2 n(e). |> e Vn 2 n(e). n+: A. I and II only B. I only C. II and III only D. I and III only E None of the above choices A, B, C or D..

Q: 3) For which a does > Σ 1 converge? Justify your answer. n(In n)a n=2

A:

Q: Which of the following mfinite senes are convergent? Select all that apply. 100 100+50+25+... 1-D0 2…

A: Which of the following infinite series are convergent

Q: Show by example that E(a,/b,) may diverge even though Ea, and Eb, converge and no b, equals 0.

A:

Q: In(n + 9) State whether the sequence ɑn = converges and, if it does, find the limit. n а) converges…

A: Given data is : an=lnn+9n

Q: Consider the sequence of functions {fn}=2 where fn: [a, b] → R defined for n 2 2 by if 0<x s -n² (x…

A: Definition: Suppose fn is a sequence of functions fn:S→ℝ and f:S→ℝ. Then fn→f pointwise on S if…

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: option  (d)  is correct

Q: Write the first five terms of p_j= j/(j+1) and then prove p_j converges to 1. Be sure to explicitly…

A: We have to solve an inequality for apsilon and j.

Q: Find the interval of convergence for Σ (k + 2) æ(k+2) k + 3 а) O1-2, 2) b) О (-1,1) с) O1-1, 1) d) O…

A: Put k+2 = t  We get  ∑(t xt )/(t+1)

Q: Which of the following inequalities can be used to show that +1 n=1 converges? 1 O n3 +1 nº + 1 O n°…

A: The series ∑n=1∞nn3+1. We have to find which of the following inequality can be used to show the…

Q: Ean converges and an > 0 for all n , can anything be said about E? An

A: If ∑ an converges and an &gt; 0 for all n then ∑ 1an  converges         Proof :…

Q: Show that if an < o0, then E In(1+an) converges if and only if an converges. n=1 n=1 n=1 Hint: show…

A:

Q: The sequence ɑn = 1 + converges to the limit A) e B) ve C) Ve D) e? E) F) e? /2

A:

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: The sequence {r"} converges: if and only if 0 srs1 а. if and only if –1<rs1 Ob. if and only if r| s1…

A: To find the value of r for which the sequence rn converges.

Q: Suppose that X₁, X2, ka)=P(X = _k) =1/2. Let Yn = Xn = n 2 Xi. are independent, such that, for each…

A: Chebyshev's Inequality Chebyshev's inequality guarantees that for a large class of probability…

Q: Suppose that an(x – 6)" converges for x = 10. At which of the points (a)-(d) must it also converge?…

A: Given: The given series is ∑an(x-6)n  The series converges at x=10. According to theorem is the…

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

A:

Q: Consider the sequence n²r(1 – nx) x€ (0, 1/n] =: (x)"f elsewhere . Show it converges uniformly on…

A: Consider the sequence  fnx=n2x1-nx      ,x∈0,1n0                       ,elsewhere The limit function…

Q: 4. Consider the 2nd to 5th entries of the following two sequences and answer the questions be- low:…

A: D-Alembert's test: Let un is a sequence .…

Q: What are all of the values of x ,such that - (-1) n+A converges? n=1 .1 X26 .2 06

A:

Q: Find the interval of convergence for k°(x – 9)* 2k a) O[-1, 1] b) O (7, 11) c) O (-1,1) d) O [7, 11)…

A: Find interval of convergence.

Q: Suppose that an<0 for all n, and define b,=tan-(a,-n). Then the sequence {b,}... O diverges to - O…

A: Solve the following

Q: ht A: If S 2n – 2n²x x € [0, 1] fx,(æ) = { otherwise converges in distribution as n → 0. nt B: If 1…

A: Given that Xn : n≥1 is an infinite sequence of random variables and for each n, the density function…

Q: For which of the sets below, does the property X, ES for all n > 53 imply that (x,) is convergent? O…

A:

Q: 13.D. Show that, if we define fn on R by fa (x) 1+ n?x? then (fn) converges on R.

A: givenfn(x)=nx1+n2x2dividing numerator and denominator by x2fn(x)=nxx21+n2x2x2

Q: The following data is given on a subset of the real line: (a) Using the distance function: d(rı, I2)…

A: The following are the steps for K-means clustering algorithm. K points from the data are…

Q: 5. (a) Suppose 1 and Xn+1 = 2 Show that {n} converges and find the limit. X1 for all nEN.

A:

Q: 25. Let a, := (), n2 1. Which of these conditions must be met if a, is to converge n+1 to 1? I for…

A: Note that for ε→0, 1ε-1→∞, thus from the condition n&gt;1ε-1, n→∞. Thus, the condition I is…

Q: 1 3) For which a does converge? Justify your answer. n(In n)e n=2

A: This question is related to series.

Q: The sequence {(1 + 1)^² +1} 2n O O converges to e² is divergent ez converges to e 2 O converges to e

A: Sequence

Q: .8.8 Suppose that Cn >0 for each n E N. Prove that n either converges = diverges to infinity.

A:

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

A:

Q: Q 12.11 whot values of does this Seirics seines converge? n (in u) n=2

A:

Q: 42. Given that la converges, does In (n) 1-1 (n²+5) ² $-10 converge or diverge? A. Dive B. Cannoe…

A:

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

A:

Q: By examining the limit as n→0, the sequence {2"}D=1 O Diverges O Converges to 0 1 Converges to

A:

Q: (4) Suppose xo = 2√√3, yo = 3, 2xn-19n-1 Xn-1+Yn-1 and Prove that (a) xnx and yn ↑y as n→ ∞o for…

A: Hints:  It is given that, x0=23, y0=3 and xn=2xn-1yn-1xn-1+yn-1 and yn=xnyn-1 for all n∈ℕ.…

Q: Suppose that uz is a series of positive terms and that g is a continuous function that is no k21.…

A: Here we  have given that, ∑k=1∞uk is the series of positive terms and that g is continuous function…

Q: Given that " In (1) converges, does [.. converge or diverge? In(n) 1 (²+5)² A. Diverges B. Cannot be…

A: Let ∑pn and ∑qn are two series such that pn≤qn. If ∑qn is convergent, then ∑pn is convergent.

Q: Show by example that E(a,/b,) may converge to something other than A /B even when A = equals 0. Σαν…

A:

Q: State whether the sequence an = Vn +7 converges and, if it does, find the limit. a) O converges to…

A:

Q: (a) Prove that 4280 k!pk k=0 converges for any r > 0.

A:

Q: B: To what value the following sequences can converge: Vn+3 %3D 1 4n+3 -Inn 2n2+1 1) an= 2) a,…

A:

Q: Q3) The sequence a, = 1 - (0.2)" a) converges to o b) diverges c) converges to e d) converges to 1

A: Given an=1-0.2n

Q: The sequence {C+1 In=2)"} N=1 converges to e² O None diverges converges to le converges to e³

A: The given sequence 3n+23n+1nn=1∞ The sequence can be written as follows:…

Q: 5) Find all values of x for which En=1 nxn (n+1)(2x+1)n converges absolutely.

A:

Q: an converges for some large k, then so will n=1 an- If E n=k

A: We have to check

Q: Find the interval of convergence for 3kzk (k + 4)² а) O1-3, 3] 1 1 b) O c) O[-4, 4] d) O(-1,1) e) O…

A: