Re: [分析] 均勻收斂
※ 引述《wyob (Go Dolphins)》之銘言：
: f_n(x)=(nx)/(1+nx^2)
: 請問x在[0,1]和在[1/π,2]這兩個區間內都是均勻收斂嗎??
: 常看到這些題目，所以把它弄懂
: 感謝
0 , x=0
lim f_n(x) =
n→inf 1/x , x€(0,inf)
(pf：1.x=0： trivial
2.x€(0,inf)：
nx x
(nx)/(1+nx^2)=─────=─────
1+nx^2 1 + x^2
─
n
when n goes to inf, the limit will be x/x^2 = 1/x)
Problem 1. x€[0,1]
We use the thereom：
At (C[a,b], supnorm)
if
1.fn(x) € (C[a,b], supnorm)
oo
2.(fn(x)) are uniformly Cauchy in (C[a,b], supnorm)
n=1
then
by Denoting lim fn(x) = f(x) (because in (C[a,b], supnorm) will be complete)
n→oo
we have
4.fn(x) converges to f(x) uniformly
5.f(x) € (C[a,b], supnorm)
---------------------------
Assume fn(x) satisfies 2.
Since fn(x) satisfies 1.
By the theorem
we have 5., leading to contradiction ( f(x) is not continuous at 0)
so fn(x) is not uni. Cauchy
absolutely not uni. conv.
Problem 2. x€[1/pi,2]
Consider
│fn(x)-f(x)│
=│ nx 1 │
│─── - ─ │
1+nx^2 x
1 1
=│────│ <= │──────│---*
x(1+nx^2) π(1+nπ^2)
Since * conv. to 0
for all epsilon .......................blablabla
Done~
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02/22 19:06, , 1F
02/22 19:06, 1F
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