MATH2023 Analysis – Semester 2, 2024 – Assignment
Due: Friday, 11th October 2024, 11:59pm
All solutions need to be justified.
The assignment must be submitted electronically as a single PDF file in Canvas. You may sub-
mit scanned copies of handwritten solutions or typeset your work. Note that your assignment
will not be marked if it is illegible or poorly scanned or submitted sideways or upside down. It
is your responsibility to check that your assignment has been submitted correctly.
Question 1. For each of the following power series, find the radius of convergence R. For
parts (a) and (b), determine if the series is conditionally convergent, absolutely convergent, or
divergent for z = R and z = −R.
(a)
∞∑
n=0
√
2nzn.
(b)
∞∑
n=1
nα(2z)n, α ∈ R.
(c)
∞∑
n=0
anz
n, where an =

−1/3n if n = 3k for some k ∈ N
5n if n = 3k + 1 for some k ∈ N
i/nn if n = 3k + 2 for some k ∈ N
Question 2. Consider the sequence (an) given by a1 = α, an+1 = an(2− an) for n ≥ 1,where
α ∈ R.
(a) Find all the values of α ∈ R such that (an) converges, and for each of these values of α,
find the limit lim
n→∞
an. (As part of your answer, you should find the values of α such that
(an) diverges.)
(b) 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].
Question 3. Let g(x) = x3 + 3x for x ∈ R.
(a) Show that the equation
g(x) = 1 + sinx
has at least one solution.
(b) Is the solution unique?
Question 4. Let fn : [0, 2]→ R and gn : [1, 3]→ R be two sequences of functions converging
uniformly to limits f and g respectively.
(a) Find the domain for functions fn + gn.
(b) Show that the sequence fn + gn is uniformly convergent to f + g on the domain found in
(a).
END OF ASSIGNMENT
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