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Unformatted text preview: MATH 104, ﬁnal test, Dec 16th.
Name
Student ID #
All the necessary work to justify an answer and all the necessary steps of a proof must be
shown clearly to obtain full credit. Partial credit may be given but only for signiﬁcant
progress towards a solution. Show all relevant work in logical sequence and indicate all
answers clearly. Cross out all work you do not wish considered. Books and notes are allowed.
Calculators, computers, cell phones, pagers and similar devices are not allowed during the
test.
1. (10pts total) Construct, with proof, a compact set of real numbers whose limit points
form a countable set. 1 2. (10pts total) Suppose that the coeﬃcients of the power series n an z n are integers,
inﬁnitely many of which are distinct from zero. Prove that the radius of convergence is at
most 1. 2 3. (10pts total, 5pts each subitem) Evaluate
(a)
(b) 1
n→∞ n n ej/n , lim 1
n→∞ n j=1
n (−1)j lim j=1 3 j
n 100 . 4. (10pts total) A realvalued function f deﬁned on an interval (a, b) is said to be convex if
f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y)
whenever a < x, y < b, 0 < λ < 1. Prove that every convex function is continuous. 4 5. (10pts total, 5pts each subitem) Suppose f is a realvalued diﬀerentiable function on IR.
Call x a ﬁxed point of f if f (x) = x.
(a) If f (t) = 1 for all t ∈ IR, prove that f has at most one ﬁxed point.
(b) Show that the function f (t) := t + (1 + et )−1 has no ﬁxed points, although 0 < f (t) < 1
for all t ∈ IR. 5 6. (10pts total, 5pts each subitem) For x real and n ∈ IN, let
fn (x) := x
.
1 + nx2 Show
(a) that (fn ) converges uniformly to a function f , and
(b) the equation
f (x) = lim fn (x)
n→∞ is correct for x = 0 but false for x = 0. 6 ...
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This document was uploaded on 11/17/2010.
 Spring '09
 Math

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