设f(x)在[0,1]设f x 在 a b 上连续续,且f(x)>0,则lim(n趋向于无穷)根号下(f(1/n)*f(2/n)*f(3/n)...f(n/n

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设f是[0,1]上的连续函数,证明lim(n趋向于正无穷)n∫(从0到1)x^nf(x)dx=f(1)
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题目没有问题∫{0,1}xⁿ*f(x)dx=∫{0,1-1/√n}xⁿ*f(x)dx+∫{1-1/√n,1}xⁿ*f(x)dx由于f(x)在[0,1]上连续,xⁿ在[0,1]上不变号,且在[0,1]上可积对f(x)在[0,1-1/√n]上运用积分第一中值定理,存在一点ξ₁∈[0,1-1/√n],使得∫{0,1-1/√n}xⁿ*f(x)dx=f(ξ₁)*∫{0,1-1/√n}xⁿdx=f(ξ₁)*[x^(n+1)/(n+1)]| {0,1-1/√n}=f(ξ₁)*(1-1/√n)^(n+1)/(n+1)对f(x)在[1-1/√n,1]上运用积分第一中值定理,存在一点ξ₂∈[1-1/√n,1],使得∫{1-1/√n,1}xⁿ*f(x)dx=f(ξ₂)*∫{1-1/√n,1}xⁿdx=f(ξ₂)*[1/(n+1)-(1-1/√n)^(n+1)/(n+1)]=f(ξ₂)/(n+1)[1-(1-1/√n)^(n+1)]故lim{n→∞}n*∫{0,1}xⁿ*f(x)dx=lim{n→∞}n*f(ξ₁)*(1-1/√n)^(n+1)/(n+1)+lim{n→∞}n* f(ξ₂)/(n+1)[1-(1-1/√n)^(n+1)]由于lim{n→∞}(1-1/√n)^(n+1)=lim{n→∞}(1-1/√n)^n=lim{x→0+}(1-x)^(1/x²)=lim{x→0+}e^[1/x²*ln(1-x)]=e^{lim{x→0+}[1/x²*(-x)]} a→0,ln(1+a)~a=0故lim{n→∞}n*f(ξ₁)*(1-1/√n)^(n+1)/(n+1)=lim{n→∞}f(ξ₁)*lim{n→∞}[n/(n+1)]*lim{n→∞}(1-1/√n)^(n+1)=lim{n→∞}f(ξ₁)*1*0 =0注:∵f(x)在[0,1]上连续,∴f(x)有界,∴lim{n→∞}f(ξ₁)为有限值∵当n→∞时,1-1/√n→1,∴ξ₂→1故lim{n→∞} n* f(ξ₂)/(n+1)[1-(1-1/√n)^(n+1)]=lim{n→∞}f(ξ₂)*lim{n→∞}[n/(n+1)]*lim{n→∞}[1-(1-1/√n)^(n+1)]=lim{ξ₂→1}f(ξ₂)*1*(1-0)=f(1) 由连续性因此,lim{n→∞}n*∫{0,1}xⁿ*f(x)dx=f(1),证毕本题若直接根据积分中值定理,得到存在一点ξ∈(0,1),使得∫{0,1}xⁿ*f(x)dx=ξⁿ*f(ξ),这里0
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题目错了吧。根据积分中值定理,式子左边等于c^n*f(c),c属于(0,1),f是闭区间上连续,所以有界,c^n*f(c)极限就是0,不是f(1)
题目确实是这样的,正是因为这样才不会啊
如果你确定这就是原题,那题目错了。
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(高数)关于极限的两道题.两题都是求x-&0时的极限1.lim(f(a+x)+f(a-x)-2f(a))/x^2 = f''(a)2.(((f(a+x)-f(a)/x-f'(a))/x = 1/2f''(a)关于第一题,但是求第二题的时候,,为什么不是这样,而是?第一种求法哪里错了啊?
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第一种方法化简的时候那个分母是f’(a)
而不是f‘(a+x)
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Here ist ein Kroki der situatuion:
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