Question

计算:1/[2(跟号1)+根号2]+1/[3(根号2)+2(根号3)]+……+1/[9（根号8）+8（根号9）].

Answer 1

1/[2√1+√2]+ 1/[3√2 +2√3]+..+1/[9√8+8√9]

先观察其通项公式,an=1/[(n+1)√n +n√(n+1)]=(1/√n)-[1/(√(n+1))]

∴S=1/1 - 1/√2 + 1/√2 -1/√3 +...+1/√8 -1/√9

=1-1/3

=2/3