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If S(n)=1+1/2+1/3+…+1/n(ninN), then prov...

If `S_(n)=1+1/2+1/3+…+1/n(ninN)`, then prove that
`S_(1)+S_(2)+..+S_((n-1))=(nS((n))-n)or(nS((n-1))-n+1)`

Text Solution

Verified by Experts

`S_((1))+S_((2))+..+S_((n-1))`
1+
`1+1/2+`
`1+1/2+1/3`
………………..
………………..
`1+1/2+1/3+………1/(n-1)`
Adding vertically
`=(n-1)+((n-2))/2+((n-3))/3+….((n-(n-1))/((n-1)))`
`=n[1+1/2+1/3+…….+1/(n-1)]-[1+1+1+….1]`
`=nS_((n-1))-(n-1)=nS_(n)-n`
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