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Material Type: Exam; Class: ADVANCED CALCULUS; Subject: Mathematics; University: University of Pennsylvania; Term: Spring 2009;
Typology: Exams
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(1) Is the following true for all complex x?
n ∑
j=
(1 + x)
j
j
xj
2
n ∑
j=
|1 + x|
2 j
n ∑
j=
j
x
2
(2) Suppose {sn} is a convergent sequence of complex numbers and {snk }
is a subsequence. Must {snk } converge?
(3) What does limn→∞
n
n equal?
(a) 0
(b) 1
(c) e.
(d) ∞.
(4) Suppose
|ai| converges and
ai = 2. Is there a rearrangement aik
of the terms such that
ai k
(5) Is it possible to have a function which is not continuous anywhere?
(6) If f (x), g(x) : R → R are continuous functions must f (g(x)) be contin-
uous?
(7) Suppose {fn} is a sequence of continuous functions on [a, b] such that
(∀x ∈ [a, b])(∀n ∈ N)|fn(x)| ≤ Mn and
Mn converges. Must
fn
converge uniformly to
f?
(8) If f
2 (x) is integrable on [a, b] must f be integrable on [a, b]?
(9) If A is disconnected must the closure of A be disconnected?
