Question

求极限lim(x→无穷)1/n{(1+cosπ/n)^(1/2)+....+(1+cosn*π/n)^(1/2)} 要过程...

Answer 1

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) dx
=(2√2/π)∫[0→1] cos(πx/2) d(πx/2)
=(2√2/π)sin(πx/2) |[0→1]
=2√2/π

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