Suppose  f is continuous on [0, ∞) and limx→∞ f (x) = 1 . Is itpossible that  ∫ 0∞f (x) dx is convergent?

Q: Give an example of a sequence of continuous functions: fn : [0, 1] →R that converges pointwise to 0…

A: Let us consider the function fn(x)=xn on [0,1]. Let u assume that the function fn(x)=xn is pointwise…

Q: 1. Use the definition of the definite integral as the limit of a Riemann sum to evaluate - 4)² dx.

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Use series to evaluate the limit. lim 1- cos(5x) x-01+ 5x - e5x

A: Objective is to find the limit

Q: Only one of these calculations is correct to evaluate the definite integral xe-X dx Select one: lim…

A: Explanation of the answer is as follows

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: Is Uniform convergence of {fn(x)} is sufficient but not necessary to transmit continuity from the…

A: From the theorem stated in step two, we can see that the function is uniformly continuous and we can…

Q: 1. Use the definition of the definite integral as the limit of a Riemann sum to evaluate

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Q: 6. Let (x,) be properly divergent and let (y,) be such that lim(x,y,) belongs to R. Show that (y,)…

A: To show that (yn) converges to 0, given (xn) is properly divergent and (yn) is such that lim(xnyn)…

Q: If lim =c>0 , then b, , a, both convergent and both divergent.

A: By limit ratio test we know that if above condition is true then either both series diverges and…

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 the sequence f.(x)} converges to a limit function (that you should determine) for each XEI…

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Q: 2. Express each definite integral as a limit using summation notation: In x dx = L(8x – x²)dx =

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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: Evaluate (x2 – 2x³) dx using the limit of Riemann sums.

A: given integration is                                       I=∫01x2-2x3dx we have to find value of…

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: Use series to evaluate the limits

A: Given limit: limy→0y-tan-1yy3 Series for tan-1y, tan-1y=∑n=0∞-1ny2n+12n+1=y-y33+y55-y77+....

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

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Q: If the function f is continuous on [0,00) and | f(x)dx is convergent, then there is a possibility…

A: If the function fIs continuous on [0, ∞) and ∫1∞f(x)dx is covergent, then there  is a possiblility…

Q: If f is defined on [0, +inf] and Lebesgue integreable, uniformly continuous function, then, lim_(x…

A: Given If f is defined on [0, +inf] and Lebesgue integrable, uniformly continuous function, then,…

Q: Show that the sequence of functions {f.} where f,(x)=: 1+ nx „xER is non-uniformly convergent on…

A: Show that the sequence of functions fn(x), where  fnx=x1+nx2,  x∈ℝ is non-uniformly convergent on…

Q: 12. Evaluate the limit n Σ lim -> sin(- n→∞ N k=1 by expressing it as an integral.

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Q: the sequence {fn} converges uniformly to f on R. Prove that the limit function - R be bounded.…

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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: Give an example of a sequence of continuous functions: fn: [0, 1] R that is pointwise convergent but…

A: Consider the function fx=0, x is irrational in 0,1mn, x is rational in 0,1. This function is called…

Q: 2 Q/if the function fc)=ĕ is not Continuous at T where =TIL x<T foint the fourier Scries of fcxeě is…

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Q: Let fn(x) x + n a. Show that fn(x) converges pointwise [0, 0). on

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

A: As per our company guidelines, we are supposed to answer only 1st question at a time. Kindly repost…

Q: Let x1 O and xn+1 x+1 2 Show that (x,n) converges and find its limit.

A: It is given that  x1=0 and xn+1=xn2+12. Let xn+1=ϕ(xn)=xn2+12 By fixed point iteration method…

Q: Use series to evaluate the limits

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Q: lim as x approaching 0 (1-cosx)/(1+x-e^x) use series to evaluate the limits

A: Given,          limx→01-cosx1+x-ex

Q: e lim エ→e | e 1.

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Q: Let fn: R --> R be defined by : fn(x)= x/(1+nx2), For all n >= 1. a) Show that {fn} converges…

A: step:-1 Theorem:- Let fn be a sequence of functions on I such that for each n∈ℕ, f'n(x) exists for…

Q: If lim=c>0, then both convergent and both divergent. ylgn O Ihi O

A: If limn→∞anbn=c&gt;0, then ∑bn,  ∑an both convergent and both divergent. (a) True (b) false

Q: Let h: [0,1] R be a continuous function with h(1) by fn(x) = x"h(x) for x e [0,1]. Show that {fn}…

A: A sequence of real-valued functions considers two types of convergence. These are pointwise…

Q: 2. Show that following sequence of functions are convergence pointwise to its limit function. a)…

A: See the attachment

Q: Find an example of a sequence of continuous functions {fn} defined on the interval [0, 1] such that…

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Q: Find the interval of convergence ofS f(x) dx where f (x) = (-1)+(x-6)" %3D E

A: Given problem is :

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: Use series to evaluate the limit. 2 +2 cos t-2 lim t+0

A: Given,           limt→0t2+2cost-2t4

Q: Evaluate the definite integral: (2 – a²)dx using limit of the Riemann sum.

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Q: Construct a sequence of continuous functions fn : R →→ R such that 0 < fn S 1, limn-o Jo fndx = 0…

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

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Q: Let f : [0, 1] → R be a continuous function. Show that the sequence of functions 1 fn (x) = ,x +1 +…

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Q: Let f(x) 2.x – 5 if x > 5 --- Calculate the following limits. Enter "DNE" if the limit does not…

A: Here we have, f(x)=10-x-x2 if x≤52x-5         if x&gt;5 We have to calculate,…

Q: Show that the sequence f.(x)3 converges to a limit function (that you should determine) for each xEL…

A: Consider the given function.

Q: is continuous lim x f(x) = 0 = f is continuous in 0.

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Q: Use series to evaluate the limit. 1- cos(5x) lim x → 0 1 + 5x – e5x

A: Authoring guidelines: As per the authoring guidelines we can solve only the first question.Please…

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.