化简根号[1+1/n2+1/(n+1)2]
1个回答

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)