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(math203)[2007](f)final~ma_yxf^_10477.pdf
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Math 203 Final Exam
December 14, 2006 Your Name
Student Number
1.
You can use and quote anything from my lecture note and the answer to my home-work.
2.
Please feel free to raise your questions.
Number Score
1
2
3
4
5
6
Total
(1) (10 pints) Suppose f(x) is continuous on an open interval containing [a, b]. Prove that
1
the limit limn fx += f(x) is uniform.
n
(2) (20 pints) Suppose f(x) is continuous and positive on [0, 1]. Prove that
12 n . 1
lim n ff f
n0 nn n
converges and .nd the limit.
(3) (20 pints) Let [x] be the biggest integer x. Let a> 0. Determine the convergence
. 1 .. ...
a 1
of the improper integral . a dx.
xx
0
(4) (20 pints) Determine the intervals on which the series nxxn uniformly converge.
. 1 . 1
(5)
(20 pints) Suppose absolutely converges. Prove that converges to a
an x . an function that has derivatives of any order away from all an.
(6)
(10 pints) True or false. No explanation needed.
1.
If an and bn converge, then anbn converges.
2.
If an and bn absolutely converge, then anbn absolutely converges.
3.
If an and bn conditionally converge, then anbn conditionally converges.
4.
If both an and one rearrangement ank converge but have di.erent sum, then
an conditionally converge.
5.
If all rearrangements of an converge, then an absolutely converges.
6.
If an converges, then max{a2n,a2n+1} and min{a2n,a2n+1} converge.
7.
If max{a2n,a2n+1} and min{a2n,a2n+1} converge, then an converges.
8.
If un(x) uniformly converges on X and g(Y ) . X, then un(g(y)) uniformly converges on Y .
9.
If un(x) uniformly converges on X and g is uniformly continuous on Y , then
g(un(x)) uniformly converges on X.
10. If fn(x) and gn(x) uniformly converge to f(x) and g(x), and fn(x) > 0, f(x) > 0, then fn(x)gn(x) uniformly converges to f(x)g(x).
Answer to Math 203 Final, Autumn 2007
not absolutely guaranteed to be correct
(1) By assumption, f(x) is continuous on [a . , b + ] for some > 0. Then f(x) is uniformly continuous on [a . , b + ]. For any .> 0, there is > 0, such that
x, y [a . , b + ], |x . y| < =.|f(x) . f(y)| < ..
Then
111 1
x [a, b],n > =. x + ,x [a . , b + ], < =. ..fx + . f(x)..< ..
n n n
1
This shows that the limit limn fx += f(x) is uniform.
. n
12 n . 1
(2) Let an = n ff f . Then
nn n
... .. ...
11 2 n . 11
log an = log f + log f + + log f = S(Pn,f) . f(0),
nnnn n
where S(Pn,f) is the Riemann sum with respect to the special partition
12 1 n . 1
Pn :0 <<< << < 1
nn nn
i . 1
of [0, 1] and xi . = . The function log f is still continuous on the bounded interval [0, 1]
n
. 1
and is therefore Riemann integrable. Therefore we have limn S(Pn,f) = log f(x)dx,
0
so that
. 1
lim log an = lim S(Pn,f) = log f(x)dx.
n n
0
Finally, by the co