Lebesgue Measures, Integrals and Limits : Let f_n(x) = n^1/2 * x * e^(-n*x^3), for n = 1,2,3... (i) Find the maximum value assumed by f_n in the interval [0,1] and (ii) Find Lim (n -> infinity) of...

Let f_n(x) = n^1/2 * x * e^(-n*x^3), for n = 1,2,3... (i) Find the maximum value assumed by f_n in the interval [0,1]. (ii) Find Lim (n -> infinity) of integral from 0 to 1 of (f_n(x))dx. All integrals here are with respect to Lebesgue measure. Please justify every step and claim. e here is the exponential function.

Lebesgue Measures, Limits and Integrals : Prove - If f is in L^1[0,1], then limit the integral over [0,1] of x^n*f = 0 as n goes to infinity.

Prove or disprove the following: If f is in L^1[0,1], then limit the integral over [0,1] of x^n*f = 0 as n goes to infinity. I saw a similar example asking to prove that the integral from 0 to 1 of x^2n f(x) dx = 0, and they used algebra of functions generated by {1,x^2}, but we haven't talked about that, so please when yo ...continues

Uniform Convergence

Please see the attached file for the fully formatted problems. We are using the book Methods of Real Analysis by Richard R. Goldberg.

Sequences and Uniform Convergence : Let {fn} infinity-->n-1 be a sequence of continuous real-valued functions that converges uniformly on the closed bounded interval [a, b]. For each nЄ I let ...

Let {fn} infinity-->n-1 be a sequence of continuous real-valued functions that converges uniformly on the closed bounded interval [a, b]. For each nЄ I let Fn(x) = ∫ x--> a fn(t)dt an-1 converges uniformly on [a,b]. (Hint: Use 9.2F) Theorem 9.2F; ...continues

Sequences of Continuous Function, Uniform Convergence and Pointwise Convergence

Let {fn(x)} n-1 ---> infinity be a sequence of continuous functions [0,1] that converges uniformly. a) Show that there exists M>0 such that |fn(x)|<= M (nЄI 0

limit

(See attached file for full problem description with equation and proper symbols) --- 9.2-10 If be a sequence of functions that converges uniformly to the continuous function , prove that ---

show that does not converge uniformly on [0,1], even though converges pointwise.

(See attached file for full problem description with proper equations) --- 9.3-3 Let . Use the result of exercise 4 of Section 9.1 to show that does not converge uniformly on [0,1], even though converges pointwise. ---

93.4

(See attached file for full problem description with proper equations) --- 9.3-4 Let . Show that converges uniformly to 0 on [0,1], but that does not converge (even) pointwise to 0 on [0,1 ---

93.5

(See attached file for full problem description with proper equations) --- 9.3-5 Let be a sequence of functions on [a,b] such that exists for every and (1) converges for some (2) converges uniformly on [a,b]. Prove that converges uniformly on [a,b].Show how this result may be used to weaken that hypothesis ...continues

94.2

(See attached file for full problem description with equations) --- 9.4-2 Does the series converge uniformly on (Hint: Find the sum of the series for all x) --- We are using the book of Methods of Real Analysis by Richard Goldberg.