已知,n >0,化简根号下1+1/n²＋1/(n+1）²
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1+1/n^2+1/（n+1）^2
={[n(n+1)]^2 +(n+1)^2 +n^2}/ [n(n+1)]^2
={(n^2 +1)*(n+1)^2 +n^2}/ [n(n+1)]^2
={(n^2 +1)*(n^2 +2n+1)^2 +n^2}/ [n(n+1)]^2
令 n^2 +1=a
则 原式可化为
={a*(a+2n)+n^2} / [n(n+1)]^2
={a^2 +2an +n^2} / [n(n+1)]^2
={a+n}^2 / [n(n+1)]^2
所以 根号[1 +1/n^2 +1/(n+1)^2]
=(a+n)/n(n+1)
=(n^2 +n+1)/n(n+1)
={[n(n+1)]^2 +(n+1)^2 +n^2}/ [n(n+1)]^2
={(n^2 +1)*(n+1)^2 +n^2}/ [n(n+1)]^2
={(n^2 +1)*(n^2 +2n+1)^2 +n^2}/ [n(n+1)]^2
令 n^2 +1=a
则 原式可化为
={a*(a+2n)+n^2} / [n(n+1)]^2
={a^2 +2an +n^2} / [n(n+1)]^2
={a+n}^2 / [n(n+1)]^2
所以 根号[1 +1/n^2 +1/(n+1)^2]
=(a+n)/n(n+1)
=(n^2 +n+1)/n(n+1)
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