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Unformatted text preview: sin 2x dx
x
2x
0
w
2 w Ý Þ sin 2x dx
lim
2x
0 and the substitution u 2x yields Þ0 2w lim
wÝ
b. Þ0 Ý sin 2 x dx
x2 Þ0 w lim wÝ c. Þ0 Ý 2 sin 2 x cos 2 x dx
x2 sin u du
u 2 sin x cos x dx lim
x
wÝ Ý w Þ0 Ý Ý sin x dx.
x sin 2 x dx lim sin 2 w
w
wÝ
x2 2 sin 2 x cos 2 x dx lim
wÝ
x2 Þ0 Þ0 Ý sin 2 x dx.
x2 Þ0 wÝ
lim
d. Þ0 Þ0 Þ0 w 2 sin 2 x cos 2 x dx
x2
w
2
x
w Ý 2 Þ sin 22 dx
lim
0
2x 2 sin 4 x dx
x2 357 2w sin 2 u du
u2 Þ0 Ý sin 2 x dx.
x2 Ý 2 sin 4 x dx
x2 Þ0 Ý Þ0 Ý Þ0 Þ0 Ý 2 sin 2 x 1 cos 2 x
dx
x2
2 sin 2 x dx Þ Ý 2 sin 2 x cos 2 x dx
0
x2
x2
Ý
2
2
2 sin x dx Þ
sin x dx Þ Ý sin 2 x dx.
2
0
0
x
x2
x2 14 Sequences and Series of Functions
Exercises on Convergence of Sequences of Functions
1. For each of the following definitions of the function f n on the interval 0, 1 prove that the sequence
f n converges pointwise to the function 0 on 0, 1 and determine whether the sequence converges
boundedly and whether it converges uniformly. In each case, determine whether or not we have Þ 0 f n 0.
1 lim nÝ In each case, use Scientific Notebook to sketch some graphs of the given function and ask yourself whether
your conclusion is compatible with what you see in the graphs.
a. f n x nx exp nx for each x 0, 1 and each positive integer n. Hint: To see that the sequence converges boundedly but not uniformly to the constant 0, observe
that each function f n has its maximum when
n exp nx n 2 x exp nx 0
which occurs when x 1/n. Note that f n 1/n 1/e for each n.
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0 0.2 0.4 x 0.6 0.8 1 When x 0 the condition
lim nx exp nx 0 nÝ is obvious and when x 0 the condition
lim nx exp nx 0
nÝ follows from the fact that
lim u 0.
eu
converges pointwise and boundedly but not uniformly to the
uÝ Therefore the sequence f n
constant function 0.
Finally we observe that Þ 0 f n x dx Þ 0 nxe nx dx
1 as n 1 Ý. 358 1
n 1
en 1
ne n 0 b. f n x n 2 x exp nx for each x 0, 1 and each positive integer n.
5
4
3
2
1 0 0.2 0.4 x 0.6 0.8 1 Arguing as we did in Part a, we see that each function f n has its maximum value n at the
e
number 1 . Since sup f n Ý as n Ý we see that f n fails to converge boundedly.
n
When x 0 the condition
lim n 2 x exp nx 0
nÝ
is obvious and when x 0 the condition
2x2
lim n 2 x exp nx 1 n Ý n nx 0
x lim e nÝ follows from the fact that
2
lim uu 0.
e
converges pointwise to the constant function 0. Finally we observe
uÝ Therefore the sequence f n
that Þ 0 f n x dx Þ 0 n 2 xe nx dx 1
1 as n 1 n
1
1
en
en
converges pointwise to the constant function 0 we Ý. So even though the sequence f n do not have Þ0 fn Þ0 0
1 lim nÝ c. f n x nx exp n 2 x 2 for each x 1 0, 1 and each positive integer n.
0.4 0.3 0.2 0.1 0 0.2 0.4 x 0.6 0.8 1 For each x and n we have
f n x n exp n 2 x 2
and so the equation f n x 0 holds when x 2n 3 x 2 exp n 2 x 2
1
2n . Since 1
1.
2e
2n
From this observation we see that the sequence f n co...
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 Fall '08
 STAFF
 Math, Calculus

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