代写R编程设计、代做R代写、代写fixed constant留学生

日期：2019-10-03 10:42

Question 1. For each of the following series, determine if they are conditionally convergent,

absolutely convergent, or divergent.

Question 2. For each of the following power series, find the radius of convergence R and

determine if it is conditionally convergent, absolutely convergent, or divergent for z = R and z = −R.

Question 3. Let fn(x) = nxe−nx2

, for x ∈ R and n ∈ N.

(a) Calculate f(x) = limn→∞

fn(x) and determine the domain of f(x).

(b) Is (fn(x)) uniformly converging to f(x) on [−1, 1]?

(c) Is (fn(x)) uniformly converging to f(x) on [1, +∞)?

Question 4. The sequence (bn) is given by: b1 = β, bn+1 = bn(2−bn), for n ∈ N, where β ∈ R

is a fixed constant.

(a) For β ∈ [0, 2], prove that the sequence (bn) is convergent and find its limit.

(b) For β ∈ (−∞, 0) ∪ (2, +∞), prove that the sequence (bn) is divergent and find lim sup n→∞bn and lim infn→∞bn.

(c) Let f : R → R be a continuous function such that f(x) = f(x(2 − x)), for each x ∈ R.

Prove that f is constant on [0, 2].

END OF ASSIGNMENT