Find the sum `3/2-5/6+7/(18)-9/(54)+oodot`

Answer

`15/16`

Explanation

`S=3/2-5/6+7/18-9/54+..oo`
`rArrS=3(1/2)+5(-1/6)+7(1/18)+9(-1/54)+..` (1)
`rArr-1/3S=3(-1/6)+5(1/18)+9(-1/54)+..` (2)
Subtracting (2) from (1), we have
`4/3S=3(1/2)+[2(-1/6)+2(1/18)+2(-1/54)+…oo]`
`=3/2+(2(-1/6))/(1-(-1/3))`
`=3/2-(1/3)/(4/3)`
`=3/2-1/4`
`=5/4`
`rArrS=15/16`