.Q. Consider (R, M(R), µ). For x E [2,3], consider the sequence of functions tan ((x - 2)"т) fn (2) = ,then fn(x) converges to 3x + 2 а. 0 а.е. b. За + 2 1 C. За +2 d. 0 a.e e. None of these.

Q: For x21, the continuous function g is decreasing and positive. A portion of the graph of g is shown…

A: We are given the graph of g to be the following.           For x≥1, the continuous function g is…

Q: The sequence {( an+2 )"}1 n=1 O diverges converges to Je3 converges to e None converges to Je?

A: the given sequence converges to

Q: The sequence V(n + 5)(n + 7) – n} is A) Convergent to 0. B) Divergent. C) limn-»0 V (n + 5)(n + 7) –…

A:

Q: Consider the sequence of functions {fn}=2 where fn: [a, b] → R defined for n 2 2 by if 0<x s -n² (x…

A: Definition: Suppose fn is a sequence of functions fn:S→ℝ and f:S→ℝ. Then fn→f pointwise on S if…

Q: (x + n)² Show that h„(x) converges poimtwise to f(x) = 1, V x e [1, 00). %3| x² + n²

A:

Q: Let f and g be as in the figure, we may deduce that if | g(x)dx converges then | f(x)dx a a | 4 y =…

A: Given two functions f(x) and g(x) It can be clearly seen that the area bounded by the function g(x)…

Q: Suppose that X₁, X2, ka)=P(X = _k) =1/2. Let Yn = Xn = n 2 Xi. are independent, such that, for each…

A: Chebyshev's Inequality Chebyshev's inequality guarantees that for a large class of probability…

Q: Consider the sequence n²r(1 – nx) x€ (0, 1/n] =: (x)"f elsewhere . Show it converges uniformly on…

A: Consider the sequence  fnx=n2x1-nx      ,x∈0,1n0                       ,elsewhere The limit function…

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

A: True

Q: Let r be a constant. Suppose >,r i=1 converges to 8, then the sum of >r' is i=0 Answer:

A:

Q: Suppose that an<0 for all n, and define b,=tan-(a,-n). Then the sequence {b,}... O diverges to - O…

A: Solve the following

Q: ht A: If S 2n – 2n²x x € [0, 1] fx,(æ) = { otherwise converges in distribution as n → 0. nt B: If 1…

A: Given that Xn : n≥1 is an infinite sequence of random variables and for each n, the density function…

Q: If 2 a, converges, then la,| diverges. n=1 n=1 True False

A: Given statement is FALSE. I HAVE given a counter -example.

Q: For each n ≥ 1, let fn : R → R, fn(x) =nx/(n + x^2) Show that the sequence of functions {fn}n≥1…

A:

Q: Solve Σ 0 to ∞ ((n! • x^n)/ n^n) for x = e and x = -e and explain whether each point converges or…

A: Solve for x = e and check whether it converges or diverges as follows.

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

A:

Q: 2. s) Let x1 = 1 and define rn+1 = v3+2x, for n E N. Show that the sequence (xn) converges and find…

A:

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

A: Since solve the first question for you. If youwant any specific question to be solved then please…

Q: 1.1 Determine whether the sequence (Jn(x)} = {, on I = [0,1]. %3D I7 converges uniformly

A: As per our norms we have to write one question at a time kindly repost remaining questions.

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

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: 4. Let r be a positive number. Determine the value(s) of r, if any, such that converges. As n" n=1…

A: Explanation of the answer is as follows

Q: Which of the followings are false? 1. E1 sin converges by comparing with II. E1 T%3D1 nl 00 n-1…

A:

Q: For which z's does the infinite series 1+ 좋 + + +. zn + . 3" 27 converge? To what function does it…

A: The solution are next step

Q: The sequence {(1 + 1)^² +1} 2n O O converges to e² is divergent ez converges to e 2 O converges to e

A: Sequence

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

A:

Q: 3n 14 Determine the rate of convergence of the sequence an sin as n → 0. A. O(1/n^2) B. O(1/n^3) C.…

A:

Q: 5. Let h, : [0, o∞) → R be given by h„(x) and let h : [0, ∞) → R be given by I +n %3D h(x) = = x.…

A: Answer

Q: (c) Given f: R → R, suppose that f is differentiable and |f' (x)| < 1 for all a E R. Show that the…

A: Please check step 2 for the solution.!

Q: The sequence {G 3n+2 n=1 O None converges to e O diverges converges to Ve? converges to e3

A:

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

A:

Q: We define an arithmetic sequence of functions: fn: [1,0) → R recursively as: fi(x) = Vx, Vx € [1, 0)…

A: Given fn:[1,∞)→ℝ recursively as: f1x=xfn+1x=x+fnx for all x∈[1,∞). (a). Let x∈[1,∞), therefore:…

Q: (b) Consider the set U = {(x, y) E R² : 2x + y < 1, x 2 0, y 2 0} and let Zn = (Xn, yn), where Xn =…

A: NOTE:- I HAVE REQUIRED QUESTION A TO SOLVE QUESTION C AND D. THEREFORE I AM SOLVING ONLY QUESTION B.

Q: For some constant a, the following improper integral | (x2 – 1)e2xdx converges to converges to e2a…

A:

Q: For n e N let fn: [0, 1] → R be the function fn(x) = e²-n. Decide whether the sequence (fn)converges…

A:

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: 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: Set an where r E R is a real number. Prove that ɑn converges to 0 if (-1, 1), cônverges tó 1 if r =…

A:

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

A:

Q: Consider the sequence S n²r(1 – nx) x € [0, 1/n] - fn(x) := 0. elsewhere . Show it converges…

A: Let gx=n2x1-nx Differentiating wrt x, we get  g'x=n2-2n3x Putting g'x=0, we get  n2-2n3x=0⇒x=12n

Q: State whether the sequence an = Vn +7 converges and, if it does, find the limit. a) O converges to…

A:

Q: The sequence {G1)}m1 5n+3 5n+1 n=1 Diverges converges to Je3 converges to Je² converges to 0…

A:

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

A: Given an=1-0.2n

Q: The sequence {C+1 In=2)"} N=1 converges to e² O None diverges converges to le converges to e³

A: The given sequence 3n+23n+1nn=1∞ The sequence can be written as follows:…

Q: Suppose fn : [0, 1] → R be defined as f,(x) = xn , for n e Z+, for x E [0, 1]. Define: f(x) := { 0,…

A: Given that fn(x)=xn (a) Define, f(x)=0x∈[0,1)1x=1 Take fn(x)=1xn. We say that fn→f pointwise if…

Q: The sequence a, n – vn+ lvn+3 converges to the limit A) -3 B) -1 C) 1 D) +2 E) -2

A:

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

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.