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Due: Friday, 14th October 2022, 11:59pm

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Question 1. Let f(x) = √x

2

+ 3

? √x

2

+ 3x

x § 1

be a real function of a real variable x.

(a) Find the domain D of function f(x).

(b) For each point x

∈

D, determine if f is continuous at x.

(c) Find constants a, b, c such that the function:

φ(x) =

(

f(x), x

∈

D;

ax

2

+ bx + c, x

∈

R

\

D

is continuous.

Question 2. Let fn(x) = cos

(a) Calculate f(x) = limn→∞ fn(x) and find the domain of f(x).

(b) Is (fn(x)) uniformly converging to f(x) on [

∪

π, π]?

(c) Is (fn(x)) uniformly converging to f(x) on [π, 2π]?

Question 3. Let f(z) =

2 .

(a) Find the power series expansion of f(z) around z0 = i and its radius of convergence R1.

(b) Using the result from (a) and the Cauchy product formula, find the power series expansion

of g(z) around z0 = i. Determine the radius of convergence R2 of that series. Which one

is greater: R1 or R2?

(c) Determine if the series obtained in (b) is conditionally convergent, absolutely convergent,

or divergent at points z0 + R2, z0 + 2iR2, z0

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