Question

Prove that:for all n>=1, 1/1.2+1/2.3+1/3.4+.......+1/n (n+1)=n/(n+1)

Answer 1

Here,
n ≥ 1

1/1.2 + 1/2.3 + 1/3.4 + .............. + 1/n(n +1)

here , we use concept of Sigma ,

Sigma 1/r(r + 1) { r = 1 to n }

Sigma [ 1/r - 1/(r + 1) ] { r = 1 to n }

put r = 1
1/1 - 1/(1 + 1) = 1 - 1/2
put r = 2
1/2 - 1/(2 +1) = 1/2 - 1/3
put r = 3
1/3 - 1/(3 +1) = 1/3 - 1/4

..............................
.............................
put r = n
1/n - 1/( n +1)

========================
after adding

Sigma [ 1/r - 1/(r +1) ] {r = 1 to n } = 1 -1/2 + 1/2 -1/3 + 1/3 - 1/4 +............1/n - 1/(n +1)

= 1 - 1/( n +1) = n/( n +1)

hence,
1/1.2 + 1/2.3 + 1/3.4 +..........1/n( n +1) = n/( n +1)

hence proved ///