※ 引述《BitTorrent (BT)》之銘言：
: Suppose a_k in R and |a_k|<=k for every positive integer k.
: ∞
: Let f(x) = Σa_k*x^k and f_n(x)=f(x+1/n).
: k=1
: show that f_n converges uniformly to f on every interval [a,b] 包含 (-1,1)
Let c = max{|a|,|b|}
∵ (|a_k|)^1/k -> 1 as k->∞
∴ f(x) converges unif. on [-c+ζ,c-ζ] for some positive number ζ< 1-c
For every ε>0, there is an integer N' such that
n'
|Σa_k*x^k|≦ε/3 for all n',m'>N'
m'
N
∵ h(x) = Σa_k*x^k is unif. conti. on [-1,1]
1
∴ there is a δ>0 such that
|h(x)-h(y)|≦ε/3 for all 0<|x-y|<δ
Choose N > max{1/2δ,ζ}, then
∞ ∞
|f_n(x) - f_m(x)|≦|h(x+1/m)-h(x+1/n)|+|Σa_k*(x+1/m)^k|+|Σa_k*(x+1/n)^k|≦ε
N+1 N+1
for all n,m>N, x in [-c,c].
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