Exercise 3. Find f(x) = lim,0 fn(x) on S. Show that (fn)n converges uniformly to f on closed subsets of S. - fn(x) = x" sin(nx), S= (-1,1). |

Q: Let x1 > 1 and  xn+1 = 2 − 1 / xn for n ≥ 2.  Show that ( xn ) converges and find its limit.

A: Given - x1>1 and xn+1=2-1xn for n≥2 To prove - xn is convergent and also find its limit .

Q: Suppose fn:A→ Ris uniformly continuous on A, Vn E N and fn →f converges uniformly on A. Question:…

A:

Q: Suppose f: R→R is a continuous, 2n – periodic function. Define the function f,(x) := f(x+ ±) for x E…

A: Given Information -  f is a continuous function on R and is 2π periodic.

Q: the series E k converges uniformly on D(0;r) for every r E (0,1), but diverges on C\ D(0; 1). More…

A:

Q: Let g : [0, 1] → R be continuous and g(1) = 0. Define gn(x) = g(x)x" for x E [0, 1] and n = 1,2,…

A: As per our guidelines we are supposed to answer only one question.

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: 3. Let { fn} be a sequence of functions such that lim fn (x)=f (x) V xE [a, b]. 我→ Let Mn=sup { fn…

A:

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: 8. Let f : R → R be uniformly continuous on R and if fn(x) = f(x + ) for x E R, show that (fn)…

A:

Q: Let fn(x) on [0, 1]. 1+ nx (a) Does fn converge uniformly on [0, 1]? (b) Check if lim | fn(x) dæ lim…

A:

Q: Exercise 3. Find f(x) = limn+00 fn(x) on S. Show that (fn)n converges uniformly to f on closed…

A: To find limn→∞fnx=fx on S where S=x:x≠±1 When x=0,  fnx=1limn→∞fnx=1 When x∈-1,0∪0,1, limn→∞x2n=0.…

Q: 11. Suppose that f is uniformly continuous on [0, M) for all M > 0 and lim, f(x) true if we do not…

A: The complete solutions are given below

Q: Express the limit as a definite integral               n lim         sigma  e^(x_i)/(1+x_i)*deltax…

A:

Q: Suppose that the sequence of functions {fn} and the function f are integrable on [a, b] and that…

A: Given that,  The functions fn and f are integrable on a,b. fn converges to f on uniformly a,b. an…

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

A: We have to check

Q: • If it converges, what is its limit? Provide some justification for your answer. • If it diverges,…

A: key points By monotone convergence theorem ;  if a sequence is decreasing and is bounded below by an…

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

A:

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

A:

Q: Let (fn) be a sequence of differentiable functions defined on the closed interval [a, b], and assume…

A:

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

A:

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: 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: Exercise 2. Determine whether the sequence of functions (fn)n converges uniformly on S. fn(x) = nx"…

A:

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: Let (fn) be a sequence of functions that are integrable on [a, b] and that converge uniformly on [a,…

A:

Q: For n e N, define fn : [-1, 1] → [0, 1] by fn(x) = (1 – x2)". For all 8 e (0,1), define Is = [-1,…

A:

Q: (f) V = {x € R: In converges as noo}, W = R, T(x) = limn → In, I EV; (g) V =R%, W = R, T(x) = -1n,…

A:

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

A:

Q: Consider the sequence of functions n+ cos(x) 3n + sin(x) defined on the R. 1. Find the function f to…

A:

Q: 1. For r < 1, define fn(x) = ²x. (a) Prove that for x < 1, k=1 ƒ(x) = lim_ƒn(x) = 84x x(1+x) (1-x)³…

A:

Q: Let f be a positive decreasing function defined on [a, +co] such that f(x) →0 as x→ + o. Let a be…

A: Consider the given positive decreasing function f defined on the interval a, +∞ such that fx→0 as…

Q: Let fn(x) = nxe-n, for x = [0, 1]. (a) Show that for each x € [0, 1] we have fn(x) converges to 0.…

A:

Q: Using the Nth term test for divergence, 00 n 5n-2 n2 n=1 diverges since lim V5n-2 n² is equal to O 5…

A:

Q: 1 Given f : [2, 0) → R, Prove that fn(x) = converges uniformly to f(x) = 0. 1+ x"

A:

Q: Let fn E C[0, 1]. Suppose that fn (x) converge uniformly on [0, 1]. 1. Show that (fn) is uniformly…

A:

Q: Exercise 1. Express the limit as a definite integral on the given interval a) lim T? cosxi Li=1' Ax,…

A:

Q: Suppose that a sequence of differentiable functions {fn} converges pointwise to a function f on an…

A:

Q: Determine the convergence of the integral tan x d.x. E (In(cos r))2

A:

Q: If(2) -f(w)| < j-비, Va,yER. a. Show that f is a continuous function on R. b. Let zo = 1, z, = f(Tn…

A: Solution.

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

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

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

A:

Q: 8x ,t = 0, 1, 2,... 7+x (a) Use the stability criterion to characterize the stability of the…

A: The given recursion is as follows: xt+1=8xt27+xt2 The equation will be:…

Q: 5. Let (n) be a uniformly bounded sequence of functions which are Riemann integrable on [a, b, and…

A: By the hypothesis there exists an real number M such that Supremum of x belongs to [a,b] of…

Q: Applying the limit comparison test, show that the series 1.1 1 2n diverges.

A: The given series is: 12+14+16+...+12n+... The series ca be written as: ∑n=1∞12n

Q: Exercise 1. Find the Taylor expansion of the function f(x) = sin x³ around the point x = 0 up to the…

A: Introduction: One of the most significant application of differential calculus is Taylor series…

Q: Let fn : (0, 1) –→ R, fn(x) := x¬" sin(x"), n E N.Which of the following statements are true? Select…

A:

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

A:

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.