Đáp án:
Xét giá trị:
`1/((n+1)sqrtn+nsqrt(n+1))`
`=1/{sqrtnsqrt(n+1)(sqrt(n+1)+sqrtn))`
`=(sqrt(n+1)-sqrtn)/{sqrtnsqrt(n+1)[(sqrt(n+1))^2-(sqrtn)^2]}`
`={sqrt(n+1)-sqrtn}/{sqrtnsqrt(n+1)}`
`=1/sqrtn-1/sqrt(n+1)`
Áp dụng ta được:
`1/(2+sqrt2) + 1/(3sqrt2+2sqrt3) +....+ 1/(2019sqrt2018+2018sqrt2019)`
`=1/sqrt1-1/sqrt2+1/sqrt2-1/sqrt3+1/sqrt3-1/sqrt4+...+1/sqrt2018-1/sqrt2019`
`=1-1/sqrt2019=(sqrt2019-1)/sqrt2019`
