a) Show by example that E( an /b.) may diverge even thagh E an and E bn converge and no b, equals 0.

Q: dt converges. There is no value of r such that (3t)-" 14.

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Q: 1 (x + 1)2 dx -2

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Q: Prove that if r >1 then f r" dx converges to 1

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Q: a) Determine if Eº, absolutely converges, conditionally converges, or diverges. ´n=2 n-yn

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Q: Suppose that |f(r)|I b where and g(r)dx converges. Prove |f(x)dr converges. that

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Q: Determine if +∞o 5 dx √x-1 is convergent or divergent.

A: Solve the improper integral then using expansion at infinity solve limit

Q: Theorem 64. Let {pk}1 converge to point If y # x, then {pk}1 does not converge to y. x.

A: Given that sequence pkk=1∞ converges to point x.

Q: 1 (a) Use the Integral Test to show that > converges. 2. n' +1 d 1 n = 0 -1 tan dx Hint: 1+x?

A: As per our guidelines, we are supposed to solve only one question.  Kindly post other questions  as…

Q: 00 Determine whether the improper integral ²+37+1 e dx converges or diverges. Show all your steps.

A: Since there is infinity in the upper bound, this is improper integral of type 1.   To evaluate an…

Q: 5. Let ni < n2 < n3... be positive integers, and let E = x E [0, 27]; sup converges. Show that E has…

A: Given :  n1≤n2≤n3 ...  be the positive integers and E be the set defined as ;                E =…

Q: If |fn(x)| < an for all r and E=1 an converges, prove that En=1 fn(1) converges uniformly.

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Q: (-1)ªn n² + 1 mula converges to a value or diverges withou

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Q: Let fn (x) uniformly converge to f(x) on [a, b]. If fn(x) E R[a, b], then f(x) E R[a, b] True False

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Q: Σ does not converge uniformly on R. k! k=0 Prove that

A: We know that ∑k=0∞xkk!=ex. So, we will show that ex does not converge uniformly on ℝ.  Let us assume…

Q: Let fn (æ) = a" – x2", defined on [0, 1] %3D is pointwise converges to f (x) = 0, on [0, 1] then ||…

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Q: z . Prove that fn (z) = sin is uniformly convergent to 0 on D(0,2).

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Q: 1/x e 7 dx y3 -1

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Q: Which of the following are convergent or divergent? Show your work. 72 i. En=2 lnn ii. En=1 3 1+ Vn…

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Q: Let fn(x) = x¹(1-x), r [0, 1] and n € N. Show that fn = 0 on [0, 1] but that {f} is not uniformly…

A: Given fn(x) =xn(1-x),             x∈[0,1] and n∈N To prove: {f'n} is not uniformly convergent.

Q: 4. Determine whether the function g(x) = roo sin²x converges or diverges by comparing it to f(x)…

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Q: (b) Prove that the improper integral ]] 8888 converges. dr dy (1 + x² + y2)3/2

A: Solution: Given improper integral, ∫-∞∞∫-∞∞dxdy1+x2+y232 As we know the convergence of the series is…

Q: let E an3 be a segence defined recurs ively by An=1 , anti=Jl+5an, n=l,2,3. Suppose that Does { 9n3…

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Q: If f(x) ≤ g(x) for all x in [a, +∞o) and converges. •+∞ f(x) da converges, then [*** g(x) dx also

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Q: a. Let a,ber with a<b, let f: [a,b]→R be twice continuously differentiable, and assume that there…

A: From the given statements option "d"  is true statement according to the equations...

Q: sin(²) En=1 is 18 √n a. 1 b. 12/12 c. Converge d. Diverge d O C O a O b O

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Q: Compute the point-wise limit of (fn) and determine if they converge uniformly on given intervals. 1.…

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Q: 10. Determine whether the given dx convergent or convergent? a. converges to o Page. b. diverges to…

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Q: A. converges B. diverges

A: Given that the series: A. ∑n=1∞15n B. ∑n=1∞n+6n2+12n+4 To find check the series converges or…

Q: Let fn(x) on [0, 1]. 1+ x" (a) Prove that fn converges uniformly to 0 on [0, €], Ve E (0, 1). (b)…

A: According to our guideline, we can solve only first question of multiple question. If you want…

Q: Σ Suppose that ax" converges to a function y such that y" – 2y' + y = 0 where y(0) = 0 and y'(0) =…

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Q: Determine whether r{(1+E)*"}.. Converges or diverges.

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Q: k+ln k > ak, where aj = and ak+1 ak k+10 %3D k=1

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Q: Suppose f(x) > 0 and g(x) > 0 and that | f(x)dx diverges. If f(x) > g(x), what can we conclude about…

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Q: Let (Jn)neN be a sequence of functions f: R → R satisfying |fn+1(c) – fn(c)| 2 and neN , and ae(0,…

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Q: A) diverges B) converges

A: If the limit exists and is a finite number, we say the improper integral converges . If the limit is…

Q: Prove with an e – N argument that {fn(x) + gn(x)} converges uniformly to f (x) + g(x) on R.

A: The objective here is to prove that : fnx+gnx converges uniformly to fx+gx Given: fnxconverges…

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 =…

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

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Q: There is no value of r such that Jo (3t)-r 5 dt converges. 14.

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Q: If fn : R → [0, 0) is a sequence of integrable functions, fn + f a.e. as n → 0, and S fn + 0. Prove…

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

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Q: b) Use the Comparison Theorem with f(x) = - to show whether , g(x)dx is convergent or divergent.

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

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Q: Determine the values of p for which f(x)da converges where f(x) = = 1-e-z TP

A: Here, in the question for what value of p the integral is convergent. The simple improper integral…

Q: Does E 8. k=1 In(ek41) converge or diverge?

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Q: If |fn(x)| < an for all x and Ln=1 an converges, prove tha En-1 fn(x) converges uniformly. n=1

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Q: Consider the sequence of functions {fn}n=2 where fn: [a, b] → R defined for n > 2 by if 0 sxs fa =…

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Q: Suppose that fn(x) and f(x) are functions R –→ R. Suppose also that fn converges uniformly to f as n…

A: Suppose that fnx and fx are functions from ℝ to ℝ such that fn converges uniformly to f as n→∞.  To…

Q: Perform a testto determine if. S a J y converges diverges or

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