( 10001 100 ! 2 1 !+5 2 !+10 3 !+ +10001 100 ! = )

Step-by-step solution

Here, general term of denominator is ( aligned & _ n=1 ¹⁰⁰ (n^2+1 ) n != _ n=1 ¹⁰⁰ [(n+1)^2 n !-2 n n ! ] & = _ n=1 ¹⁰⁰[(n+1)(n+1) !-n n !]- _ n=1 ¹⁰⁰ n n ! & = _ n=1 ¹⁰⁰[(n+1)(n+1) !-n n !]- _ n=1 ¹⁰⁰(n+1) !-n ! & [ array ccc 2 2 ! & - & 1 1 ! +3 3 ! & - & 2 2 ! & & +101 101 ! & -100 100 ! array ]- [ array cc 2 !- & 1 ! +3 !- & 2 ! +101 !-100 ! array ] & =[101 101 !-1]-[101 !-1] aligned ) Denominator (=100 101 ! ) Consider, ( 10001 100 ! 100 101 100 ! = 10001 10100 ) Hence, option (b) is correct.