求极限lim(x→无穷)1/n{(1+cosπ/n)^(1/2)+.+(1+cosn*π/n)^(1/2)} ..
问题描述:
求极限lim(x→无穷)1/n{(1+cosπ/n)^(1/2)+.+(1+cosn*π/n)^(1/2)} ..
答
lim[n→∞] (1/n)[(1+cos(π/n))^(1/2)+...+(1+cos(nπ/n))^(1/2)]=lim[n→∞] (1/n)Σ(1+cos(iπ/n))^(1/2) i=1到n=∫[0→1] [1+cos(πx)]^(1/2) dx=∫[0→1] [2cos²(πx/2)]^(1/2) dx=√2∫[0→1] cos(πx/2)...