※ 引述《plover (>//////<)》之銘言：
: ※ 引述《plover (>//////<)》之銘言：
: : Let f be a positive continuous function defined on [a,b]. And let M be the
: : maximum of f. Show that
: : b
: : ( ∫ f(x)^n dx )^1/n -> M, as n goes infinity.
: : a
: Hint: 在 f 達到最大值那一點使用連續性.
∵f is conti. on [a ,b]
exist t. in [a,b] such that f(t.)=M
any ε>0 exist δ>0 such that x in (t.-δ,t.+δ)
=> M-ε<f(X)<M+ε
t.+δ b n b
=> ∫ f(t)^n dt < ∫ f(t)^n dt < M ∫dt
t.-δ a = a
n b n
=> (2δ) (M-ε) < ∫ f(t)^n dt < M (b-a)
a =
1/n b 1/n 1/n
=> (2δ) (M-ε) < [ ∫ f(t)^n dt ] < M (b-a)
a =
∵εis arbitrary
1/n b 1/n 1/n
=> (2δ) M < [ ∫ f(t)^n dt ] < M (b-a)
a =
b 1/n
==> lim [ ∫ f(t)^n dt ] = M
n->∞ a
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◆ From: 218.184.18.190
※ 編輯: icged 來自: 218.184.18.190 (01/20 17:47)