D = $\dfrac{1}{2018}$ ( $\dfrac{2}{1}$ + $\dfrac{3}{2}$ + $\dfrac{4}{5}$ + .... + $\dfrac{2019}{2018}$
= $\dfrac{1}{2018}$ ( 2018 + 1 + $\dfrac{1}{2}$ + $\dfrac{1}{5}$ + .... + $\dfrac{1}{2018}$
`= 1 +` $\dfrac{1}{2018}$ `( 1 +` $\dfrac{1}{2}$ `+` $\dfrac{1}{5}$ `+ .... +` $\dfrac{1}{2018}$
Tách `1 +` $\dfrac{1}{2}$ `+` $\dfrac{1}{5}$ `+ .... +` $\dfrac{1}{2018}$ = C
`⇒ C = 1 +` ($\dfrac{1}{2}$ `+` $\dfrac{1}{5}$ `+ .... +` $\dfrac{1}{2018}$)
Ta có: $\dfrac{1}{2}$ `>` $\dfrac{1}{2019}$
$\dfrac{1}{3}$ `>` $\dfrac{1}{2019}$
`......`
$\dfrac{1}{2018}$ `>` $\dfrac{1}{2019}$
`⇒ 1` `+` ($\dfrac{1}{2}$ + $\dfrac{1}{5}$ `+ .... +` $\dfrac{1}{2018}$) `( 2017` số`)`
`= 1 +` $\dfrac{1}{2019}$.2017
`= 1 +` $\dfrac{2017}{2019}$
Khi đó:
`D= 1 `+ $\dfrac{1}{2018}$`.(1 +` $\dfrac{2017}{2019}$`)`
`= 1 +` $\dfrac{1}{2018}$ `+` $\dfrac{2017}{2019}$
`=` $\dfrac{2019}{2018}$ `+` $\dfrac{2017}{2019}$
