Let `S_(k) =sum_( i =0) ^(oo) (1)/((k +1)^(t)),` then `sum _(k =1) ^(n) kS_(k)` equal :

Options

A. `(n(n+1))/(2)`
B. `(n(n-1))/(2)`
C. `(n(n+2))/(2)`
D. `(n(n+3))/(2)`

Answer

D

Explanation

We have, `S_(k)=underset(ntooo)limunderset(i=0)overset(n)sum(1)/((k+1)^(i))`
`rArr" "S_(k)=1+(1)/((k+1)^(2))+(1)/((k+1)^(3))+ . . . .. `
`rArr" "S_(k)=(1)/(1-(1)/(k+1))=(k+1)/(k)`
`:." "underset(k=1)overset(n)sumkS_(k)=underset(k=1)overset(n)sum(k+1)=2+3+ . . . +(n+1)`
`rArr" "underset(k=1)overset(n)sumkS_(k)=(n)/(2)(2+n+1)=(n(n+3))/(2)`