Q .Solve (1 / 2 + 1 / 6 + 1 / 12 + 1 / 20 + 1 / 30 + 1 / n(n+1)) = ?

We can break the series as-

= 1 / 1(1+1) + 1 / 2(2+1) + 1 / 3(3+1) + 1 / 4(4+1) + 1 / 5(5+1) + 1 / n(n+1)

If we take T_n as a last term -

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

= ( 1 - 1/2 ) + ( 1/2 - 1/3 ) + ( 1/3 - 1/4 ) + ( 1/4 - 1/5 ) + ( 1/5 - 1/6 ) .......( 1/n - 1/(n+1) )

Now solve the series -

= 1 - 1 / (n+1)

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

= n / (n+1)

? = n / (n+1)