√[1+1/n²+1/(n+1)²]
=1/[n(n+1)]* √[n²(n+1)²+(n+1)²+n²]
=1/[n(n+1)]* √[n²(n+1)²+2n(n+1)+1]
=[n(n+1)+1]/[n(n+1)]
=1+1/[n(n+1)]
=1+1/n-1/(n+1)
所以
原式变为
1+1/1-1/2 + 1+ 1/2 - 1/3 +...+ 1+ 1/100 - 1/101=100+1-1/101=101-1/101
√[1+1/n²+1/(n+1)²]
=1/[n(n+1)]* √[n²(n+1)²+(n+1)²+n²]
=1/[n(n+1)]* √[n²(n+1)²+2n(n+1)+1]
=[n(n+1)+1]/[n(n+1)]
=1+1/[n(n+1)]
=1+1/n-1/(n+1)
所以
原式变为
1+1/1-1/2 + 1+ 1/2 - 1/3 +...+ 1+ 1/100 - 1/101=100+1-1/101=101-1/101