Question

Help me by step by step

Answer 1

let's do this by partial sums


`((1)/(5)-(1)/(6))+((1)/(6)-(1)/(7))+((1)/(7)-(1)/(8))+`
`((1)/(8)-(1)/(9))+((1)/(9)-(1)/(10))+...((1)/(n+1)-(1)/(n+2))` etc
we notice that there are pairs that cancel

`(1)/(5)+(-(1)/(6)+(1)/(6))+(-(1)/(7)+(1)/(7))+(-(1)/(8)+`
`(1)/(8))+(-(1)/(9)+(1)/(9))+(-(1)/(10)+...(1)/(n+1))+(-(1)/(n+2))`
we will then be left with the end terms which are

`(1)/(4+1)-(1)/( \infty+2)`

`(1)/( \infty+2)` basically simplifies to 0 because it is basically 1/infintiy

so we are left with
`(1)/(4+1)` or
`(1)/(5)`

the sum is
`(1)/(5)`