lim(1/(n^2+1)+2/(n^2+2^2)+……+n/(n^2+n^2))

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lim(1/(n^2+1)+2/(n^2+2^2)+……+n/(n^2+n^2))

lim(1/(n^2+1)+2/(n^2+2^2)+……+n/(n^2+n^2))
lim(1/(n^2+1)+2/(n^2+2^2)+……+n/(n^2+n^2))

lim(1/(n^2+1)+2/(n^2+2^2)+……+n/(n^2+n^2))
原式=lim(n->+∞)[1/(n²+1²)+2/(n²+2²)+.+n/(n²+n²)]
=lim(n->+∞){1/n[(1/n)/(1+(1/n)²)+(2/n)/(1+(2/n)²)+.+(n/n)/(1+(n/n)²)]
=∫(0,1)xdx/(1+x²) (由积分定义得,∫(0,1)表示从0到1积分)
=1/2∫(0,1)d(1+x²)/(1+x²)
=[1/2ln(1+x²)]|(0,1)
=1/2(ln2-ln1)
=ln2/2

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