Find the coefficient of x 100 in 3 - 5 x 1 - x 2 .

3 − 5 x 1 − x - 2 = 1 + 2 x + − 2 − 2 − 1 x 2 2 ! … . . + − 2 − 2 − 1 . . − 2 − 98 x 99 99 ! + − 2 − 2 − 1 … − 2 − 99 x 100 100 ! Coefficient of x 100 is 3 × 101 ! 100 ! - 5 × 100 ! 99 ! = - 197 .

Find the coefficient of x 100 in 3 - 5 x 1 - x 2 .

3 − 5 x 1 − x - 2 = 1 + 2 x + − 2 − 2 − 1 x 2 2 ! … . . + − 2 − 2 − 1 . . − 2 − 98 x 99 99 ! + − 2 − 2 − 1 … − 2 − 99 x 100 100 ! Coefficient of x 100 is 3 × 101 ! 100 ! - 5 × 100 ! 99 ! = - 197 .

Assuming x to be so small that x 2 and higher powers of x can be neglected. Show that 1 + 3 4 x - 4 16 - 3 x 1 2 8 + x 2 3 is approximately equal to 1 - 305 96 x .

As x is so small that x 2 and higher powers of x can be neglected, we can approximate 1 + x n ≈ 1 + n x Here, we have to approximate the value of 1 + 3 4 x - 4 16 - 3 x 1 2 ( 8 + x ) 2 3 = 1 + 3 4 x - 4 16 1 2 1 - 3 16 x 1 2 ( 8 ) 2 3 1 + x 8 2 3 = 1 + 3 x 4 - 4 1 - 3 x 16 1 2 1 + x 8 - 2 3 ≈ 1 + - 4 3 4 x 1 + 1 2 - 3 x 16 1 + - 2 3 x 8 ≈ 1 - 3 x 1 - 3 32 x 1 - x 12 ≈ 1 - 3 x - 3 32 x 1 - x 12 on neglecting x 2 term ≈ 1 - 99 32 x 1 - x 12 ≈ 1 - x 12 - 99 32 x On neglecting x 2 and higher powers of x ≈ 1 - 305 96 x .