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(math203)[2006](f)final~ma_yxf^_10476.pdf
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Math 203 Final Exam
December 19, 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) (15 points) Suppose f(x) is continuous on an open interval containing [a, b]. Prove that
. b
f(x + h) . f(x)
lim dx = f(b) . f(a).
h0 h
a
Can you weaken the continuity condition to merely integrable?
(2) (20 points) Suppose f(x) is integrable on [a, b]. For any partition P of [a, b] and choices xi ., de.ne the Riemann product
.. .
(P, f) = (1+ f(x1)x1)(1 + f(x2)x2) (1 + f(xn)xn).
R b
f(x)dx
a
Prove that lim.P .0 (P, f)= e .
. 1
. (.t)n.1
txdx =
(3) (15 points) Prove that x . Justify each step.
nn
0
n=1
tx tx log x.]
[Hint: expand x= e
. 1
(4) (15 points) Suppose an > 0 is increasing. Prove that converges if and only if an
. n
converges.
a1 + a2 + + an
log an
(5) (20 points) Suppose an > 0 and r = limn . Consider the series f(x)=
log n
. an n=1 .
nx
1.
Prove that for any r. >r, we have an <nr. for su.ciently big n.

2.
Prove that for any R>r + 1, the series uniformly converges on [R, +).

3.
Prove that f(x) has derivatives of any order on (r +1, +).

4.
Prove that the series diverges on (.,r). On the other hand, show that the series may converge for some x<r + 1 by constructing an example.


(6) (15 pints) True or false. No explanation needed.
1. If f(x) and g(x) are integrable on [a, b], then f(x)g(x) is integrable on [a, b].
f(x) on[a, b]
2.
If f(x) is integrable on [a, b] and g(x) is integrable on [b, c], then

g(x) on(b, c] is integrable on [a, c].

3.
If f(x)+ g(x) is integrable on [a, b], then f(x) and g(x) are integrable on [a, b].

4.
If f(x)g(x) is integrable on [a, b] and 0 <c1 f(x),g(x) c2 for some constants c1 and c2, then f(x) and g(x) are integrable on [a, b].


5.If (an + bn) converges, then an and bn converge.
6.
If an and bn absolutely converge, then (an + bn) absolutely converges.

7.
If an and bn conditionally converge, then (an + bn) conditionally converges.

8.
If an and bn diverge, then (an + bn) diverges.

9.
If fn(x)+ gn(x) does not uniformly converge, then fn(x) and gn(x) do not uniformly converge.

10.
If fn(x) and gn(x) uniformly converge, then fn(x)+ gn(x) uniformly converges.

11.
If fn(x) uniformly converges, then any subsequence fnk (x) uniformly converges.

12.
If the even subsequence f2n(x) and the odd subsequence f2n+1(x) uniformly con-verge, then fn(x) uniformly converge.

13.
If |un(x)| uniformly converges, then un(x) uniformly converges.

14.
If un(x) and vn(x) uniformly converge, then un(x)vn(x) uniformly converge.

15.
If un(x) 0 and un(x) uniformly converge, then un(x)2 uniformly converges.


Answer to Math 203 Final, Autu