已知(π/2)<θ<(3π/4),则lim (n→∞)(sin^nθ+sin^(n-1)θcosθ+sin^(n-2)θcos^2θ+...+cos^nθ)/sin^(n+1)θ
热心网友
(sin^nθ+sin^(n-1)θcosθ+sin^(n-2)θcos^2θ+...+cos^nθ)/sin^(n+1)θ=1/sinθ+cosθ/(sinθsinθ)+cos^θ/(sin^θsinθ)+...+cos^nθ/(sin^nθsinθ)=(1/sinθ)(1+ctgθ+ctg^θ+...+ctg^nθ)=(1/sinθ)(1-ctg^(n+1)θ)/(1-ctgθ)=(1-ctg^(n+1)θ)/(sinθ-cosθ)∵(π/2)<θ<(3π/4),∴0