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lim(n→无穷)[1/(n^2+1)+2/(n^2+2)+.+2n/(n^2+n)]
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1/(n^2+1)+2/(n^2+2)+.+n/(n^2+n)
≤1/(n^2+1)+2/(n^2+1)+.+n/(n^2+1)
≤(n^2+n)/2(n^2+1)····(1)
1/(n^2+1)+2/(n^2+2)+.+n/(n^2+n)
≥1/(n^2+n)+2/(n^2+n)+.+n/(n^2+n)
≥(n^2+n)/2(n^2+n)
=1/2
(1)的极限为1/2 根据夹逼定理 极限为1/2
≤1/(n^2+1)+2/(n^2+1)+.+n/(n^2+1)
≤(n^2+n)/2(n^2+1)····(1)
1/(n^2+1)+2/(n^2+2)+.+n/(n^2+n)
≥1/(n^2+n)+2/(n^2+n)+.+n/(n^2+n)
≥(n^2+n)/2(n^2+n)
=1/2
(1)的极限为1/2 根据夹逼定理 极限为1/2
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