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All solutions need to be justified. You can only quote the results
from the teaching materials available on the Canvas page of
MATH2023
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Question 1. For all α > 0 and p > 0, find the radius of convergence for the power series
∞∑
n=0
nαp
√
nzn.
Question 2. Let g : [0, 10]→ R be a continuous function, such that g(0) = 0 and g(10) = 10.
(a) Show that the equation
xg(x) = 50
has at least one solution.
(b) Assuming that g is strictly increasing, that is
x1 < x2 =⇒ g(x1) < g(x2),
show that the solution to the equation above is unique.
Question 3. Let fn(x) = sin
nx2 + π
n|x|+ 2 for each x ∈ R and n ∈ N.
(a) For every x ∈ R calculate f(x) = lim
n→∞
fn(x) and find the domain of f .
(b) Is function f continuous on its domain?
(c) Is (fn) uniformly converging to f on [−π, π]?
(d) Is (fn) uniformly converging to f on [π, 2π]?
You may use the fact that for all a, b ∈ R we have
| sin a− sin b| ≤ |a− b| .
Question 4. Let (an) be a sequence of real numbers, such that an ≥ 1 for every n ≥ 1, and
lim sup
n→∞
an = 1. Show that the sequence (an) is convergent and lim
n→∞
an = 1.