Функциональные последовательности и ряды - 27 стр.

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ЮФУ | Ростов-на-Дону

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[a, b]
lim
n→∞
b
Z
a
f
n
(x) dx =
b
Z
a
³
lim
n→∞
f
n
(x)
´
dx =
b
Z
a
f(x) dx?
(f
n
)
f
n
(x) = n
α
sin x cos
n
x, x ∈ [0,
π
2
], 0 < α ≤ 2, n ∈ N,
f(x) ≡ 0
[0,
π
2
]
f(x) = lim
n→∞
f
n
(x) = lim
n→∞
n
α
sin x cos
n
x = lim
n→∞
n
α
sin x
1
cos
n
x
=
= lim
n→∞
αn
α−1
sin x
−n(−sin x)
cos
n+1
x
= α lim
n→∞
n
α−2
cos
n+1
x ≤ α lim
n→∞
cos
n+1
x = 0.
π
2
Z
0
³
lim
n→∞
f
n
(x)
´
dx =
π
2
Z
0
f(x) dx = 0.
f
n
π
2
Z
0
f
n
(x) dx =n
α
π
2
Z
0
sin x cos
n
x dx = −n
α
π
2
Z
0
cos
n
x d (cos x) =
= − n
α
cos
n+1
x
n + 1
¯
¯
¯
¯
π
2
0
=
n
α
n + 1
.
lim
n→∞
π
2
Z
0
f
n
(x) dx =
+∞, 1 < α ≤ 2,
1, α = 1,
0, 0 < α < 1.

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