求极限1/sqrt(1^2+n^2)+1/sqrt(2^2+n^2)+.+1/sqrt(n^2+n^2)

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求极限1/sqrt(1^2+n^2)+1/sqrt(2^2+n^2)+.+1/sqrt(n^2+n^2)

求极限1/sqrt(1^2+n^2)+1/sqrt(2^2+n^2)+.+1/sqrt(n^2+n^2)
求极限1/sqrt(1^2+n^2)+1/sqrt(2^2+n^2)+.+1/sqrt(n^2+n^2)

求极限1/sqrt(1^2+n^2)+1/sqrt(2^2+n^2)+.+1/sqrt(n^2+n^2)
这道题可以把要求的极限化为一个定积分来做.
lim (n→∞) 1/sqrt(1^2+n^2)+1/sqrt(2^2+n^2)+.+1/sqrt(n^2+n^2)
= lim (n→∞) 1/n*1/sqrt((1/n)^2+1^2) +1/n*1/sqrt((2/n)^2+1^2)+.+1/n*1/sqrt((n/n)^2+1^2)
= lim (n→∞) (1/n-0/n)*1/sqrt((1/n)^2+1^2) +(2/n-1/n)*1/sqrt((2/n)^2+1^2)+.+(n/n-(n-1)/n)*1/sqrt((n/n)^2+1^2)
令f(x)=1/sqrt(x^2+1),a[k]=k/n,其中k=0,1,2...n-1,n.
则对于k

integration!!!

因为:1/sqrt(n^2+n^2)《1/sqrt(k^2+n^2)《1/sqrt(1^2+n^2)
所以:
n/sqrt(n^2+n^2)《1/sqrt(1^2+n^2)+1/sqrt(2^2+n^2)+......+1/sqrt(n^2+n^2)《n/sqrt(1^2+n^2)
limn/sqrt(n^2+n^2)=limn/sqrt(1^2+n^2)=1
由夹...

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因为:1/sqrt(n^2+n^2)《1/sqrt(k^2+n^2)《1/sqrt(1^2+n^2)
所以:
n/sqrt(n^2+n^2)《1/sqrt(1^2+n^2)+1/sqrt(2^2+n^2)+......+1/sqrt(n^2+n^2)《n/sqrt(1^2+n^2)
limn/sqrt(n^2+n^2)=limn/sqrt(1^2+n^2)=1
由夹逼定理:lim1/sqrt(1^2+n^2)+1/sqrt(2^2+n^2)+......+1/sqrt(n^2+n^2)=1

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化极限为定积分的方法与解说如下,点击放大图片: