lim(n->∞)[√(1+cosπ/n)+√（1+cos2π/n）+……+√（1+cosnπ/n）]*1/n=

Question

如题所述

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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Answer 2

简单分析一下，答案如图所示