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18.100B/C FALL 2008: PRACTICE EXAM
#2
(1) See problem set 9 for Taylor’s theorem and Taylor series.
(2) (a) Let
f
:
X
→
R
be uniformly continuous. Show that if (
a
n
) is Cauchy
in
X
, then (
f
(
a
n
)) is convergent in
R
.
(b) Conversely, suppose
f
:
X
→
R
maps Cauchy sequences to convergent
sequences. Is it necessarily true that
f
is uniformly continuous?
(3) Give examples of
(a) an absolutely convergent series
∑
∞
n
=1
a
n
with lim sup
n
→∞
ﬂ
ﬂ
ﬂ
a
n
+1
a
n
ﬂ
ﬂ
ﬂ
=
∞
;
(b) a divergents series
∑
∞
n
=1
a
n
with lim inf
n
→∞
ﬂ
ﬂ
ﬂ
a
n
+1
a
n
ﬂ
ﬂ
ﬂ
= 0 .
(4) Assume that
∑
∞
n
=1
a
n
is a convergent series and that
a
n
≥
0 for all
n
∈
N
.
(a) Prove that
∑
∞
n
=1
a
2
n
converges.
(b) (hard!) Give an example showing that
∑
∞
n
=1
na
2
n
does not necessarily
converge.
(5) (a) Let (
p
n
) be a sequence in a metric space (
X, d
), and let
a
n
=
d
(
p
n
, p
n
+1
).
Assume that the series
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 Fall '10
 Prof.KatrinWehrheim
 Taylor Series

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