夹逼准则与定积分定义相结合
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设求极限的那个式子是f(n)
那么
[1/(n+1)]*[sin(π/n)+sin(2π/n)+....sin(nπ/n)]<f(n)<(1/n)[sin(π/n)+sin(2π/n)+....sin(nπ/n)]
lim(1/n)[sin(π/n)+sin(2π/n)+....sin(nπ/n)]=lim ∑(1/n)sin(kπ/n)=∫(0->1) sin(πx)dx=2/π
lim[1/(n+1)]*[sin(π/n)+sin(2π/n)+....sin(nπ/n)]
=lim[n/(n+1)]* (1/n)[sin(π/n)+sin(2π/n)+....sin(nπ/n)]
=1*(2/π)
=2/π
所以原极限=limf(n)=2/π
那么
[1/(n+1)]*[sin(π/n)+sin(2π/n)+....sin(nπ/n)]<f(n)<(1/n)[sin(π/n)+sin(2π/n)+....sin(nπ/n)]
lim(1/n)[sin(π/n)+sin(2π/n)+....sin(nπ/n)]=lim ∑(1/n)sin(kπ/n)=∫(0->1) sin(πx)dx=2/π
lim[1/(n+1)]*[sin(π/n)+sin(2π/n)+....sin(nπ/n)]
=lim[n/(n+1)]* (1/n)[sin(π/n)+sin(2π/n)+....sin(nπ/n)]
=1*(2/π)
=2/π
所以原极限=limf(n)=2/π
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