If e^x 1-x =B₀+B₁ x+B₂ x^2+ +B_n x^n+ then B_n-B_ n-1 equals

Step-by-step solution

We have, aligned & e^x 1-x =B₀+B₁ x+B₂ x^2+ +B_n x^n+ & e^x(1-x)⁻¹=B₀+B₁ x+B₂ x^2+ +B_n x^n+ & (1+ x 1! + x^2 2! + + x^ n-1 (n-1)! + x^n n! + ) & (1+x+x^2+ +x^ n-1 +x^n+ ) & =B₀+B₁ x+B₂ x^2+ +B_n x^n+ aligned On comparing the coefficients of x^n and x^ n-1 on both sides, we get aligned & 1 n! + 1 (n-1)! + + 1 2! + 1 1! +1=B_n & aligned & and 1 (n-1)! + 1 (n-2)! + + 1 2! + 1 1! +1 &=B_ n-1 & B_n-B_ n-1 = 1 n! aligned aligned