(Review of Chapter 7 PMA) Find a sequence {fn} of continuous functions on [0, 1] such that: (a) fn(x) → 0, point-wise on [0, 1], | [₁ fn (2) dx | < S fn (x) dx < M < ∞, for all n € N, for a positive constant M. 1 (b) (c) lim n→∞ fn(x) dx = 1 0 dx = 0.

Q: 24 A step function is, by definition, a finite linear combination of characteristic functions of…

A: We will take the sequence of functions {gn} as the nth order Fourier series expansion of f(x) and…

Q: Discuss limit(n →∞)fn on B and C for  fn(x)= ∑nk=1xk; B=(-1,1); C=[-a,a], |a|<1 fn(x)=1/(1+nx);…

A:

Q: (6) The symbol dx simply indicates that x (7) The sum f(x) Ax is called the German mathematician cha…

A:

Q: 818 ..) Let f: (0, ∞) → R. Assume that lim f(x) = ∞. Prove that for every sequence (n)-1 such that…

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

Q: Find the intervals where f(x) = x²e¯x is increasing and where it is decreasing. (Select all that…

A: Graph of f(x):Inflection point:Graph of f(x):

Q: Question 3 (a) State and prove sequential criteria J (b) Show by e 8 definition that the function…

A: To Do: (a) State and prove sequential criteria of continuity. (b) Show by ε-δ definition that the…

Q: sequence of non-ne a point xo. Then for every k e N Lim %D n-00

A: If f(x0)&gt;k, then there exists a number n0∈N such that fn(x0)&gt;k for all n&gt;n0Then…

Q: Let fn f, where if 0<x<! if <a<i -x if 0 <x<1 fn (x) n-2 (2 - x) 2 and f(x) %3D if x = 0 Then lim…

A: 2

Q: П 9. In(1+ kx) sin r x(ekz – x – 1) lim \k=2

A:

Q: A relation F : A → B is a function if and only if Va E A. O |F(a)| = 1 O |F(a)| > 1 O |F(a)| > 1…

A: First option is true by definition only.

Q: (a) Suppose the functions f and g are continuous on [a, b], differentiable on (a, b), and g'(x)0 for…

A: For 1st part we will use ROLLE'S THEOREM Suppose f(x) is a real valued function in the closed…

Q: 00 00 If 0 < f(x) < g(x) for all r and | f(x)dr diverges, g(x)dx diverges. then

A:

Q: Consider the function fa : [1,2)R defined by : fa(2) = 6- 2r. The expression of the upper sums,…

A:

Q: 3. For each part, sketch a function f that satisfies all the conditions, if possible. If not…

A:

Q: 1. Define f(x) = & Prove that f: R→ R is continuous. if z 20 -2² if r ≤0

A: Given: fx=x2if x≥0-x2if x≤0 Using sequences we have to prove that f:R→R is continuous.

Q: Find (a) lim x--4- f(x), (b) lim f(x), and(c) lim f(x). X- -4+ x- -4 for x > -4 15) f(x) = ! x + 4…

A:

Q: fn(2) (a) f is continuous on [0, 00). (c) f(x) = 0 for all z ≥ 0. nx, if 0≤x≤1/n; if x > 1/n. =. Let…

A:

Q: x -9 Check the continuity of f given by f(x)%3= is 0 <x <5 and x 3 at the points 3 %3! x -2x-3 1.5

A:

Q: 1.Solve the following polynomial equations.  a)  2X3+1=x2+2X

A:

Q: Let {fn}n>1 and {9n}n≥1 be measurable functions on (S,F,μ) to R and an (w) ↑ f(w) for all w & S.…

A:

Q: Evaluate the limit. lim 318 3n+1 (n + 1)! (-1)+2. 3n nl (-1)"+1 10 16 22 (6n + 4) 10 16 22 ... .…

A:

Q: Consider the sequence 19√ √n 3-√√n n=1

A: Consider the sequence 19n3-nn=1∞

Q: Suppose that 0.3210 approximates 0.3245. How many significant digits does the approximate have? O 3…

A:

Q: Let f: [1,1] → R be continuous and even f(x) = f(-x) for all x = [-1,1]. Prove: For any € > 0 there…

A:

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

A: (a) Given xt+1=8xt27+xt2 ⋯⋯(1) , t=0,1,2,… To find the fixed points, we need to solve,…

Q: Let fn f, where if 0<<! Sn(x) ={ S 2-a if 0<x<1 and f(x) if a = 0 Then lim sup fn (x) – ƒ (x)| =

A:

Q: 2+x²- 2x Given g(x)=- 3 use the theorem to show that the fixed-point sequence will converge to the…

A: Given: g(x)=2+x3-2x3 and p0∈-1, 1.1 Theorem: If g:a, b→a, b is a continuous function, then g has…