If the function f is continuous on [0,00) and | f(x)dx is convergent, then there is a possibility that | f(x)dx can be divergent.

Q: 2. Determine if S dx is convergent or divergent. x²-x

A: In this question we have to determine if this integral convergent or divergent. If limits exists…

Q: If limy- f(x) = 0, then f(x)dx converges Select one: O True O False

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Q: Q2 Prove that the scries 2+ x2 3+X2 4+x is Unitormly convergem? in any interval?

A: The given series is: 11+x2+12+x2+13+x2+……. Therefore the given series can be written as:…

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

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Q: Show the uniformly converge to F Fn CX) = X 12 Fn (X)= X 1X€ (-10] XER %3D

A: We know that if a sequence of function has bounded derivative the uniformly convergent

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

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Q: Σ 1 sin kæ) is uniformly convergent on (-0, 00). Show that k2 k=1

A: In the given question we have to prove that the given series…

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: Suppose that, given g(x), you need to prove convergence of accompanying integral via limit…

A: Solution for 1: given ∫012sinπx2x32(1-x2)dx, g(x)=1x Let  f(x)=sinπx2x32(1-x2) So…

Q: Show that if Σ i=1 converges towards a € R , then lim ) a; = 0 i=N

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Q: 2.66 Let fn(x) = z for all r E R. a) Find the pointwise limit of fn. b) Does f converge uniformly on…

A: Convergence and uniform convergence of the sequence

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 1 Σ = 0. Vn - 1+ yn converges because lim Vn – 1+ yn 2=1 True False

A: We have to check

Q: 1 -dx xln*x 1.

A: To determine if the following is a convergent or divergent. ∫e∞1xln4xdx Solution: A indefinite…

Q: Suppose to solve for a root of f(x), i.e., ƒ(x*) = 0, we use the iteration Xk+1 = ¢(xk) where o(x)…

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Q: 2.1 Show that a sequence of Riemann-Integrable functions fn(r) converges uniformly to a…

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Q: 1 dx is convergent True 8

A: Solution is given as below:

Q: Find an example of fn → f in measure, but for every x, fn(x) does not converge to f(x).

A: We have to find an example of fn→f in measure, but for every x, fn(x) does not converge to f(x).

Q: Show that f„(x) = x" (1– x") converges pointwise V x € (-1,1]. Is f„(x) converges uniformly?

A: When a sequence includes a variable, it becomes a sequence of function. A sequence of function can…

Q: 1/x e 7 dx y3 -1

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Q: If a converges to a(z) then lim / a (2)dz= / a(z)dz n- 00 Y

A: True

Q: f fn → ƒ uniformly and the functions fn have only jump discon- sinuities, prove that f has only jump…

A: Here we use epsilon delta proof

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: 5. Use the comparison test to determine if each integral converges or diverges b) 2 2/3-4 a) e-x² dx

A: Solution: 5.a ∫1∞e-x2dx

Q: If f is a continuous function on [0, ∞0) and | f(x)dx converges, then I f(x)dx diverges. 3 Select…

A: Given that, f is a continuous function on [0,∞). And, ∫3∞f(x)dx  converges. Then,…

Q: Let S f(x) dæ = S". f(x) dx + So® f(x) dx. If S". f(x) dæ is convergent, then Select one:

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

A:

Q: Determine whether

A: Solution: Consider the given integral equation: J = ∫-11dxx1/3 Since the given integrand is defined…

Q: 1 d.x x In (ex)

A:

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: Determine if the given is convergent or divergent. (x²+1)² n Zn=1 (n²+1)²

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

A:

Q: 11. Suppose f and g are continuous positive functions. Provide an example such that diverges and…

A: We need to provide example for convergence.

Q: If {fn}n=1 is a pointwise convergent, then {fn}=1 is uniformly convergent.

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Q: If lim,f(x) = 0, then f(x)dx converges Select one: O True False

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Q: 2+cosx 1. Use the comparison Theorem to determine whether dx is convergent or divergent

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Q: [ 1+2x)e*dx is convergent

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Q: 1 -dx x In*(ex) 1. e

A: Topic = Integration If the limit exists and is a finite number, we say the improper integral…

Q: 1 and yn+l = V2 + Yn for n > 1. Show (ym) is monotone, and Let yn bounded, and compute its limit.

A: Here, we have a recurrence relation for the sequence. by using that we need to check the sequence is…

Q: Determine whether the statement is true or false. If it is true, explainwhy. If it is false, explain…

A: Given:  ∫a∞f(x)dx and ∫a∞g(x)dx are convergent. To Show: ∫a∞f(x)+g(x) dx is convergent.   We…

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

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Q: Suppose f and g are continuous functions. Provide an example such that | g(x)dr diverges 0. adf(x)dr…

A: Follow next step

Q: dx heorem 1.7.1 is convergent if and only if µ < 1. (x – a)H a

A:

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

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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: (B) Let {fn(x)}1 = {r ln x. cos(x – 1)"} be a sequence of meaurable functions defined over [1,2].…

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Q: cos" (;) is absolutely convergent, conditionally convergent, or divergent. en?

A:

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…

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.