Đáp án:
`M=2009/2010`
Giải thích các bước giải:
`M= 1/2(1 +1/(1+2)+1 / (1+2+3)+...+1/(1+2+3+ ... +2009))`
`=>2M=1 +1/(1+2)+1 / (1+2+3)+...+1/(1+2+3+ ... +2009)`
`=>`$2M=1+\dfrac{1}{3}+\dfrac16\ +\,.\!.\!.+\ \dfrac{1}{\dfrac{(2009+1).2009}{2}}$
`=>`$2M=1+\dfrac{1}{3}+\dfrac16\ +\,.\!.\!.+\ \dfrac{1}{\dfrac{2010.2009}{2}}$
`=>M=1/2+1/6+1/12+...+1/2009.2010`
`=>M=1/1.2+1/2.3+1/3.4+...+1/2009.2010`
`=>M=1-1/2+1/2-1/3+1/3-1/4+...+1/2009-1/2010`
`=>M=1-1/2010`
`=>M=2009/2010`
Vậy `M=2009/2010`.
