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Due: Friday, October 13, 2023, 11:59pm
All solutions need to be justified. You can only quote the results
from the teaching materials available on the Canvas page of
MATH2023
The assignment must be submitted electronically as a single PDF file using Turnitin in Canvas.
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correctly. Late assignments will not be marked, except under special consideration.
Question 1. For all α > 0 and p > 0, find the radius of convergence for the power series
∞X
n=0
n
α
p
√
n
z
n
.
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) = limn→∞ 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 limn→∞ an = 1. END OF ASSIGNMENT