`1+1/(1+2)+1/(1+2+3)+1/(1+2+3+n)=(2n)/(n+1)`

Explanation

Let P (n) `:1 +(1)/(1+2)+(1)/(1+2+3)+……`
`+(1)/((1+2+3+.....+n))=(2n)/(n+1)` ltbr. For n=1
`L.H.S. =1`
`R.H.S. =(2.1)/(1+1) =(2)/(2) =1`
`:. " "L.H.S.=R.H.S.`
Therefore,P (n) is true for n=1
Let P (n) true for n =K .
`P(k) :1 (1)/(1+2) +(1)/(1+2+3)+....`
`+(1)/(1+2+3+....+K)=(2K)/(K+1)`
For n =k +1
`P (k+1) =1 + (1)/(1+2)+(1)/(1+2+3)+.....+(1)/(1+2+3+.....+K)`
`+(1)/(1+2+3+.....+K+(K+1))`
`(2K)/(K+1)+(1)/(1+2+3+......+K+(K+1))`
`=(2k)/(K+1) +(1)/((K+1)(K+2))=(2K(K+2)+2)/((K+1)(K+2))`
`=(2(K^(2)+2K+1))/((k+1)(K+2)) =(2(K+1)^(2))/((K+1)(K+2))`
`(2(K+1))/(K+2)=(2(K+1))/((K+1)+1)`
`rArr` P (n) is also true for n=k+1
hence form the principle of mathematical induction P (n) is true for all natural numbers n.