{au, ) IL x y² dx dy Q4-A) Show that the sequence B) Evaluate : ah (sinn)}, +sinh( converges to (1).

Q: (b) Let a1 = 1 and an+1 = lim, an- /am+2. Prove that a, is convergent and find %3D

A: The recursive relation represent a bounded above increasing sequence

Q: Use the ratio test for sequences to prove that lim,+ n'/n? = 1.

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Q: (f) Assume that g(x) > bna" is convergent on (-R, R). Instead of assuming that f(rn) = 0 assume that…

A: We will prove the part (f) using two theorems   Theorem (a)Let f(x)=∑anxn be a non-constant power…

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

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Q: Compute the limits of the sequences sin (n² – 3n + 7) n² +1 | 2n? +п -5 (b) b, (c) c, = n°e (а) а,…

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Q: 5n 9. Use the L'Hopital's rule to determine if the sequence an = () converges or diverges as limit n…

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Q: Let the function g be defined by g(x) = (x^6 + 2)/7. Use the Fixed-Point Theorem to show that the…

A: This question is basically based on the theorem named Banach fixed point Theorem which states that:…

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: Q: Use the Squeeze theorem to estabich cennergence of the gi , ren Sequence Conr Inx X(X+2)

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Q: Let =-7 and I+(3, +4) for all integers n2 1. (a) Show that xnS 2 for all n. (b) Show that (x,} Is an…

A: All proofs are easily only  a) proved using induction  b) proved using definition c) find using a…

Q: The limit of n tan(-) as n-∞is equal to 1

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

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Q: Let Ln denote the left-endpoint sum using n subintervals. Compute L4 for f (x) = work. 1 a/x-1 on…

A: Given that :- fx=1xx-1 on [2 , 4] L4=?

Q: 1.) Prove that the sequence An= nt n In (2+1) converges to O. That is, prove limln (^+1)=0 nyo

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Q: x+k², ifx < 2 What value(s) of k will make f(x) = - continuous? %3D |kx + 10, if x 2 2

A: If the function is continuous then left limit is equal to right limit

Q: Find the point wise limit of the sequence &{fn} the sequence {fr} Where Sin nx In (x) = √n if tay J…

A: Here we use basic limit property

Q: Prove that the sequence xn has a convergent subsequence. 3n4 sin n – 4n cos n2 + 2 a) Xn 5n4 – 8n –…

A: We have to prove that xn has a convergent subsequence in each of the given cases.

Q: what is the limit of this sequence limn->infinity (cos((2n+1)pi/2)

A: Limit of the sequence

Q: what is the limit of this sequence limn->Infinity (n*sin(π/n))?

A: The objective is to find limn→∞nsinπn

Q: A)(sin x)/ is convergent or not B){x} is convergent or not

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Q: EXERCICE 7. Let be the recurrent positive sequence (аn)2 3 1 + Зan-1 avec ao = 0 a) Provide the…

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

Q: (a) Prove that (2x + 2) > 1 for all x > 0. (b) Let so > 0, and sn+1= } (2s, + ) for all n > 0. Prove…

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Q: 00 Let r be a constant. Suppose ' converges to -1, then the sum of )r' is i-1 i-0

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Q: Find the limit of the sequence an = (cos n) 4n

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Q: Let Xn=cos пл 2 then limit supremum of the sequence is

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Q: 3log2(2x + 4) +6= 15 What is the value of x in the above equation?

A: By using logarithm property, we calculate the value of 'x'.

Q: Q3) The sequence a, = 1 - (0.2)" a) converges to 0 b) diverges c) converges to e d) converges to 1

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Q: Find the limit as n -> infinity of ln((9n-7)/(6n+4))?

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Q: Use the Squeeze Theorem to find the limit of the sequence. sin() 4n sin

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Q: Find (f o g)(x) for the given functions f(x) and g(x). f(x) = 5x+9 and g(x)= -6x+3 %3D

A: f(x)=5x+9, g(x)=-6x+3

Q: Find the limit inferior and limit superior of the sequence {[5 sin n]}.

A: The given problem is to find the limit inferior and the limit superior for the given sequence of {[5…

Q: Find the exact value that the sequence sin" 1 converges to. (Note that as usual, sin" x means (sin…

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Q: Suppose the function f(x) has a unique zero P in the interval [a,b]. Further, suppose f''(x) exists…

A: Suppose the function f(x) has a unique zero P in the interval [a,b]. Further, suppose f''(x) exists…

Q: Let X = (xn) and Y = (yn) be sequences of real numbers that converge to x and y, respec- tively, and…

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Q: = n(1 cos- n -

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Q: The limit cim cos CITtht1 reeresents Jerivative of sume function flX) at Some Point X= C.what ale…

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Q: Suppose xn2 0 for all n E N and the sequence (-1)"xn converges. What is its limit?

A: A sequence represented by say bn; if converges thenlimn→∞bn=0. It is important to note that the…

Q: does the sequence an converge or diverge? Find the limit if the sequence is convergent…

A: givenan=cos(3pi2 - 6n)

Q: Let {an}n=1 be a sequence of real numbers. We say that {an} decreases without bound, n=1 written lim…

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Q: Show using sequences of functions that 2 lim e dx = 0.

A: