- 数学难题2
- 1. f'(x)=2a+b,f'(0)=2n,, b=2n.f(-4n)=0,a=1/2, ∴ f(x)=(1/2)x+2nx.
2. 1/A(n+1)=f'(1/An),1/A(n+1)-1/An=2n, (1/A2-1/A1)+(1/A3-1/A2)+…+[1/An-1/A(n-1)]=2(1+2+…+(n-1)], ∴ 1/An-1/A1=n^-n, 1/A1=1/4, ∴ 1/An=(2n-1)^/4, An=4/(2n-1)^
3. ① ∵ 4/(2k-1)^=4/[4k(k-1)+1]<1/[k(k-1)], k=2-->n,∑Ak=(1-1/2)+(1/2-1/3)+…+[1/(k-1)-1/k]=1-1/n, ∴ A1+A2+A3+…+An<4+(1-1/n)=5-1/n<5
② √(An)=2/(2n-1),√[AnA(n+1)]=4/[(2n-1)(2n+1)]=2[1/(2n-1)-1/(2n+1)], ∵ k=1--->n, ∑[1/(2k-1)-1/(2k+1)]=(1-1/3)+(1/3-1/5)+…+[1/(2n-1)-1/(2n+1)]=1-1/(2n+1), 而n≥1,∴ -1/3≤-1/(2n+1)<0, 2/3≤1-1/(2n+1)<1, ∴ k=1---->n, 4/3≤∑√[AnA(n+1)]<2