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Unformatted text preview: SelfAssessment Questions – Math 118A, Fall
2009
1. Evaluate the limit
n
lim n −
n→∞
e 1
1+
n n . without using L’Hospital’s rule.
2. Prove that n 1
1+
n lim
n→∞ exists using the following steps:
(a) Prove that
n an = 1
1+
n
1− 1
n n bn = and are both increasing sequences. For this you might want to use the
fact that
√
s1 + · · · + s m
m
s1 . . . sm . ≤
.
m
(b) Prove that
cn = 1
1+
n n+1 is a decreasing sequence.
(c) Prove that an ≤ cn , and prove that the limit exists.
3. Let {Fn } be the Fibonnaci sequence, i.e.,
F1 = 1; F2 = 2; Fn+1 = Fn + Fn−1 ∀n ≥ 2.
Prove that the series ∞ n=1 converges, and evaluate it.
1 Fn
3n 2
4. Take as a given that
∞ π2
1
=.
n2
6 n=1 Evaluate ∞ 1
,
(2n − 1)2 n=1 and ∞ n=1 (−1)n 1
.
n2 5. Call a mapping of X into Y open if f (V ) is an open set in Y whenever
X is an open set in X . Prove that every continuous open mapping
from R into R is monotonic.
6. Consider a sequence of positive numbers {an } such that
an+m ≤ an + am ∀n, m ≥ 0.
Prove that an
n→∞ n
lim exists.
7. Evaluate the following limit directly:
x n − an
,
x→a x − a
lim for n ∈ N. ...
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 Fall '09
 Garcia
 Math, Topology, Open set, Topological space, continuous open mapping

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