Sum of the series `Sigma_(r=1)^(n) rlog((r+1)/(r))` is

Options

A. `log[((n+1)^(n))/(n!)]`
B. `(log(n+1)^(n))/(n!)`
C. `n!log(n+1)`
D. None of these

Answer

A

Explanation

Using `C_(1)rarrC_(1)+C_(2)+C_(3), " we get " Delta=2|(y+z,z,y),(z+x,z+x,x),(x+y,x,x+y)|`
Using `C_(2)rarrC_(2)-C_(1),C_(3)rarrC_(3)-C_(1)`,we get
`Delta=2|(y+z,-y,-z),(z+x,0,-z),(x+y,-y,0)|=2|(0,-y,-z),(x,0,-z),(x,-y,0)|" "["Using"C_(1)rarrC_(1)+C_(2)+C_(3)]`
`=2xyz |(0,1,1),(1,0,1),(1,1,0)|=4xyz`
`"Now "4xyz=12impliesxyz=3`
The number 3 can be assigned to any of x,y,z.
Therefore, the number of positive integral solution of `Delta=12` to 3.