Функциональные последовательности и ряды - 35 стр.

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ЮФУ | Ростов-на-Дону

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[0, 1]
1
Z
0
f
n
(x) dx =
1
Z
0
nx
1 + n
2
x
2
dx =
1
2n
ln
¡
1 + n
2
x
2
¢
¯
¯
¯
¯
1
0
=
ln (1 + n
2
)
2n
,
1
Z
0
f(x) dx =
1
Z
0
0 dx = 0.
lim
n→∞
1
R
0
f
n
(x) dx
lim
n→∞
1
Z
0
f
n
(x) dx = lim
n→∞
ln (1 + n
2
)
2n
= lim
n→∞
2n
2(1 + n
2
= 0.
lim
n→∞
1
Z
0
f
n
(x) dx =
1
Z
0
f(x) dx,
f
n
[0,1]
6⇒ f
(f
n
) f
n
: [0, 1] −→ R
f
n
(x) =
ln (1 + n
2
x
2
)
2n
x ∈ [0, 1] f
n
[0, 1] f
0
n
(x) =
nx
1 + n
2
x
2
x ∈ [0, 1]
(f
0
n
) [0, 1]
g(x) = 0 (f
n
)
x = 0
f
n
→ 0 x ∈ [0, 1]
0 < x ≤ 1
lim
n→∞
f
n
(x) = lim
n→∞
ln (1 + n
2
x
2
)
2n
= lim
n→∞
2nx
2
2 (1 + n
2
x
2
)
= 0.
f(x) = lim
n→∞
f
n
(x) = 0, x ∈ [0, 1].
f(x)
0
= 0 = g(x) (f
n
)
[0, 1]
[0, 1] f
f
0
n
[0,1]
6⇒ f
0

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