The sequence an =n-Vn +1vn+3 converges to the limit A)-3 B) -1 C) 1 D) +2 E) -2 F) 0

Q: Exercise 10. Let {an}=1 be a sequence converges to a limit L E R. Prove that any subsequence of…

A: Given a sequence ann=1∞ is converges to a limit L∈ℝ  Let <bnk> be a subsequence of…

Q: |an also converges because 0 If an converges then < lan|, for n = 1, 2, ... n=1 n=1

A:

Q: By squeeze theorem, the sequence an= is n2 converges to, OA. 0 OB.1 Ов. Oc. D.

A: The given sequence lies between limn→∞cos-πn2 and limn→∞cosπn2.

Q: Let f(r) = > anx" converges on an interval (-R, R) and let rn be a non zero sequence in n=0 (-R, R)…

A:

Q: Consider the sequence {fn(x)}, where sin(nx) fn(x) = 1+ nx Use the uniform norm to show that the…

A: Given: fn(x)=sinnx1+nx To show: fn(x) is uniformly converge on [1, ∞).

Q: If 2n=1a, converges, and if a„ # 1 and a,n > 0 for all n, a. Show that E=1a, converges. b. Does…

A:

Q: Is the sequen ce in (R.To)? (0,1.0.1.0,1,..) convergent Prove the answer

A: It is given that the topology space ℝ,τ0. Here, τ0=ℝ,∅. That is, the open sets of ℝ  are ℝ and ∅.…

Q: The sequence {( an+2 )"}1 n=1 O diverges converges to Je3 converges to e None converges to Je?

A: the given sequence converges to

Q: Let xn be a sequence in a metric space X. Suppose that x, has a subsequence xn, converges to a point…

A: Given, xn  be a sequence  in a metric space X and  xn has a subsequence xnjthat converges to  a…

Q: b) Prove that the following sequence is convergent, and find their limit xk =(ke-k2,cosk Vk?+k -k¥

A:

Q: Establish the convergence or divergence of the given sequenoes in full detail: (a) In = n!. (b) %. =…

A:

Q: 1. Prove the sequence (a,} = (1+" Converges to the limit e*, what about the sequence {b,} = (1-5?…

A: Note : as per our guidelines we are supposed to answer only one question. Kindly repost other…

Q: Determine the rate of convergence of the sequence an = sin( as n + 00. A. O(1/n) B. O(n) OC.…

A: given sequence an=sin2n+3n2-5  to find the rate of convergence of sequence as n→∞

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: Using Sandwich Theorem, the sequence 1 cos? n + 1 an = 4 - n converges to 4 converges to 2 diverges…

A: Given

Q: given parameters a,b > o and the given sequence x₁= b X = n+1 4 (x₂ + 4a) n Xn a. Show that a b.…

A:

Q: The sequence a, converges to the limit A) e B) ve C) Ve D) e2 E) F) e2

A: Given : The sequence     To find the convergence of the given sequence.

Q: a. In a metric Space (X, d) if (xn) and (yn) are two sequences in X such that : ХX, — хеХапd yn —…

A:

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

A:

Q: Calculate the Radius and the Interval of Convergence na" 2n n=0

A: we will be using ratio test for the series to converge.

Q: (a) In a metric space (M,d), prove that the sequence (b) = (b, b,b,...) is converge to the point b.…

A:

Q: Find the radius of the given puer of convergence (2n)! 2n %3D n=I (n!)2

A:

Q: Consider the sequence {fn(x)}, where sin(nx) fn(x) = 1+ nx a. Show that fn(x) converges pointwise to…

A:

Q: Using Sandwich Theorem, the sequence 2 an = 4 – Cos n2 + 1 - O diverges O converges to 4 converges…

A:

Q: State the definition of a sequence rn converging to a point r in a metric space (X, d). Prove that…

A: I have given the definition of the conference sequence in metric space and that used for given…

Q: Prove that 「+ X,- ズー 2. 71 ,nyo converged number J2 ヒo

A:

Q: Suppose that the sequence {an} converges to a and that la| < 1. Prove that the sequence {(an)"}…

A:

Q: Let (x,) be a convergent sequence of real numbers and let (a,) and (b,) be subsequences of (x,) with…

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: I En=1 an converges then 2a, converges.

A:

Q: If (am) is increasing and en (an) is bounded by M converges to a, {|a1\, |a|}. max

A: Let (an) is increasing sequence and converges to a,

Q: Set an where r E R is a real number. Prove that ɑn converges to 0 if (-1, 1), cônverges tó 1 if r =…

A:

Q: If X1, Y1, are two positive unequal numbers and Xn = (xn- 1+ Yn – 1) and y, = Vx, – 1 Ya-1 V n 2 2.…

A:

Q: State the definition of a sequence xn converting to a point x in a metric space (X, d). prove that…

A:

Q: ** Consider the sequence (xn) in R, where n (-1)". Show that (rn) doesn't %3D converge to any limit.

A:

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

A:

Q: If Ea, converges absolutely, prove that Ea,² converges.

A:

Q: Given a sequence ( xn ), prove that (xn ) converges to zero if and onlyif the sequence ( |xn| )…

A: From the given information, it is needed to prove that:

Q: Show that if (sn,tn) converges to (s,t) in ℝ2 with the usual d2 metric, then the sequence converges…

A:

Q: Does a(n)= n^2sin(3/n^2) converge?  If so, to where

A: The given question can be solved as shown in step2.

Q: Given f: [2, )IR, prove that fn (x) = けx" It, converger uni formly to f6)-o.

A:

Q: 2n+1 4. Use the definition to prove that ) converges.

A:

Q: The sequence a=n-vn+1Vn+3 converges to the limit A) -3 B) -1 C) 1 D) +2 E) -2 F) 0

A: For a series ; if it converges then . If  the limit ; then the series diverges.

Q: Let {B,} C R be a sequence and x E [0,1]. Prove that if |B,,41 – B„| 1 'n+: then {B,} is Cauchy and…

A: We give the detailed solution below correcting a slight typo in the problem.

Q: Suppose {f,} is a sequence in Banach space B s.t. |[fn - fn=i|l s /2-). Show that the sequence {f,}…

A: This is a problem of Functional Analysis. Banach space.

Q: Suppose > anconverges to 8 and > b,n converges to 5. What does (Зап — 2b) сonverge n=1 n=1 to? n=1…

A:

Q: The set S on wc fn Cx) = 1+ x" Converges pointwire ?

A:

Q: The sequence a, n – vn+ lvn+3 converges to the limit A) -3 B) -1 C) 1 D) +2 E) -2

A:

Q: O Find the set s on w/c In )= x"(1-x") converges pointwise.

A:

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.