Question

How can i solve this?

Answer 1

`7/6`

Explanation:

`6/(k(k+1)(k+2)) = A/k+B/(k+1)+C/(k+3) rArr {(3A=6),(4A+3B+C=0),(A+B+C=0):}`

giving

`6/(k(k+1)(k+3)) = 2/k-3/(k+1)+1/(k+3)`

then

#{ (2/1+2/2+2/3+2/4+2/5+2/6+ cdots), (0/1-3/2-3/3-3/4-3/5-3/6- cdots), (0/1+0/2+0/3+1/4+1/5+1/6+ cdots), (2-1/2-1/3+0+0+0+cdots):}#

The last row is the addition of the former three

Concluding

`lim_(n->oo) sum_(k=1)^n6/(k(k+1)(k+2)) = 2-1/2-1/3 = 7/6`