lim(n/(n^2+1)+n/(n^2+2^2)+……+n/(n^2+n^2))
来源:百度知道 编辑:UC知道 时间:2024/07/04 18:13:56
n^2 指n的平方
lim(n/(n^2+1)+n/(n^2+2^2)+……+n/(n^2+n^2))
=lim1/n*[1/(1+1/n^2)+1/(1+(2/n)^2)+……+1/(1+(n/n)^2)]
根据定积分的定义。
对于∫1/(1+x^2) x∈(0,1)
=lim∑1/n*(1/(1+(i/n)^2)
=lim1/n*[1/(1+1/n^2)+1/(1+(2/n)^2)+……+1/(1+(n/n)^2)]
则lim(n/(n^2+1)+n/(n^2+2^2)+……+n/(n^2+n^2))
=lim1/n*[1/(1+1/n^2)+1/(1+(2/n)^2)+……+1/(1+(n/n)^2)]
=∫1/(1+x^2) x∈(0,1)
=arctanx x∈(0,1)
=π/4
极限是:pi/4.
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