Qus.3: Let {fn} be a sequence in L"() such that |fn - fmllo - Oas n, m→ 0. Then show that there exists a function f such that lim f =f a.c., f E L (4) and lim ||fn- flo - 0.

Q: Od. A function f: Rm→Rª is continuous if and only if for every x in Rm and ɛ>0, there exists 8>0…

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Q: D. Use the definition of limit of a sequnce: lim an = L = Ve > 0,3N > 0s t. whenver n> N then Ja, -…

A: We have to prove limit by given limit definition Hence  1. For a sequence 8n2n+3 Consider ε =…

Q: 1. Let {an}1 be a convergent sequence such that V5 and an+1 V5 a,. aj = Compute lim an.

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Q: (a) Use the Bernoulli inequality to show that for all n ∈ N: 1 ≤ n1/n ≤ (1+√n)2/n ≤ (1 + 1/√n)2 b)…

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Q: 3. Let f : R - {0} → R be such that the sequence (xnf (xn)) converges to 3 whenever (xn) is a…

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Q: Q22. A Sequence has been defined as the following Tan aj = 1, ɑn+1 2 + an You must find its limit,…

A: a1=1,  an+1=7an2+an

Q: COs n An n

A: Answer

Q: 04: A) Consider the sequence (f,(x) of functions f,:R→R defined by: f.x)=Cos"(x) for-a/25xSa/2.…

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Q: 00 Fn+1 { is convergent, and Show that the sequence Fn n=1 1 Fn+1 lim n→∞ Fn Fn+1 > 1 and Fn-1…

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Q: = 3. Find the pointwise limit g of the sequence {n} on I = [0, 0), where gn(x) = the E-definition…

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Q: 2. Determine whether the following sequence converges, and if so, find its limit. (a) {Vn² + 3n -…

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Q: Let 1 ≤ p < ∞ and let {fn} be a sequence of functions in Lp[a, b]. Suppose there is a function f ∈…

A: We are given a sequence of functions {fn} in Lp[a,b].

Q: Find the limit of the sequence {an} = {-Yón}. Enter as an exact value. V 4n+1 limn→o An help…

A: To find the limit of a given sequence.  {an}.

Q: (b) Find the limit L of the sequence an = n° e° n+13 + n > 1 as n tends to infinity.

A: Topic:- limit calculus

Q: -3 sin n Consider the sequence b, = . In order to apply the squeeze theorem and determine the limit…

A: To find the limit of bn-3sin2nn using squeeze theorem as n goes to infinity.

Q: 2. Suppose f : R" →R satisfies the following: whenever {r,} C R" is a sequence satisfying lim In =…

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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: Suppose that is a convergent sequence and {an} 3n a, = 12· lim 2n+1) Find lim an. O48 OB. 18 OC12…

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Q: (-1)" | * n In=1 Consider the sequence n (-1)" (A) Find two sequences {xn} and {yn} such that, and…

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Q: A. Consider the recursively defined sequence s¡ = 1 and Sn+1 = 1 / (7 - Sn) for n 2 1. i. Prove that…

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Q: 2, and %3D The sequence {an} is defined by a1 2 1 an an+1 an for n > 1. Assuming that {an}…

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Q: EE- f (n)en converges in L² metric to f. In other words, we have n=- Show that N lim f – E f(n)en =…

A: we show the given result using weirstrass approximation

Q: 1. Find the limits of the following sequences as n → ∞ if they exist (Note: n N - {0}): (a) Zn √2+ i…

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Q: Find, if it exists, the limit of the sequence {k/n}, where k is a positive constant. O A. 1 О В.2 O…

A: Given function is: y=k1nTaking log on both sides, we getlog(y)=1nlog(k)Now, taking limit of n to…

Q: Find the limit of the sequence an={sin(n)/n}..if it exists.

A: Take limit n tends to infinity for finding the limit.

Q: 15. Let s1 = 3 and Sn+1 = 6+ Sn for n >1. Show that the sequence {Sn} %3D %3D is bounded and…

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Q: Find the limit of the following sequence or determine that the sequence diverges. { cos (0.99n) +…

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Q: • Let (xn) and (Yn) be bounded sequences and xn < Yn. Show that lim inf xn < lim inf Yn . n00

A: Sequence question

Q: The sequence {an} is defined by a1 = : 2, and An + An An+1 = for n > 1. Assuming that {an}…

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

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Q: Prove the limit comparison theorem i. Suppose that ap 20 and bn20 and that an =L lim n+ Then either…

A: See the attachment

Q: 3. Consider the sequence an = sin n = 1,2, 3, 4,. Find lim sup an, lim inf an. n-00 n00

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Q: Evaluate An+1 lim for n! An nn and determine the serie Σ n=1 converge or diverge? 1/e and diverge e…

A: We have to check

Q: Assume that the sequence shown below converges and find its limit. V10, /10+ V10 , /10 + /10+ V10 ,…

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Q: Find the limit of the following sequence or determine that the sequence diverges. {(cos n)/(n)}

A: Given an=cosnn

Q: In = Vn (Vn +1- vn

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Q: (a) f(0) = 0 (b) f'(0) = 0 (c) There exists another non zero sequence in (-R, R), yn, that converges…

A: This is a multipart question. We shall solve all the three parts for you. Solving this question…

Q: 5 Set up an expression for f Inx dx as a limit of sums. 3

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Q: If fn If 0 ≤ an ≤ bn for all n € N, and f on [a, b], then lim ſå fn ‡ få f. an converges, then bn…

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Q: (e) lim Vn2 +2 -n = 0

A: Limit of the indeterminate from

Q: Let the sequence {fn(x)} be defined on [0, 1] by n – n2x; if x E (0, ) fn(x) : otherwise SHOW THAT:…

A: The given sequence is fnx=n-n2xif x∈0,1n0otherwise First to find the limit function of this sequence…

Q: Use the appropriate limit laws and theorems to determine the limit of the sequence. d, In (n² + 5) –…

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Q: Find the limit of the following sequence or determine that the limit does not exist. {n sin()}

A: Solve the following

Q: Consider the Function. Sequences (fr), (gn) and (hu) given for nem* by fa: R → R÷x + x² 2 J √√x ² +…

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Q: Consider the sequence {a,}1. Suppose lim a, exists and equals L. Prove lim an+1 L.

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Q: Let f : R- (0} R be such that the sequence (rnf(rn)) converges to 3 whenever (rn) is a sequence of…

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Q: Use series to evaluate the limits

A: Given: To find:  lim t→0  1- cos t - (t22)t4Formula:  cos t…

Q: 2. Let r,LERand f:(-0,r) → R be a function. Prove that lim f(x) =L if and only if, for every X--…

A: Let r,l∈ℝ and f: -∞,r→ℝ is a function. If limx→-∞f(x)=L then for every sequence xn in -∞,r such that…

Q: Applying the ratio test to the series ∞ 1 k you would compute ak+1 lim = lim k→∞ k→∞ ak k=1 11

A: Given query is to apply ratio test.

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.