If $\frac{e^x}{1-x}=B_0+B_{1}x+B_2{x}^2+\cdots +B_n{x}_n+\cdots$ , then $B_n-B_{n-1}$ is equal to 

$\frac{e^x}{1-x}=B_0+B_{1}x+B_2{x}^2+\cdots +B_n{x}_n+\cdots$

L.H.S $={(1-x)}^{-1}{e}^x$

$=(1+x+x^2+x^3+\cdots )(1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots )$

$B_n=\text{coefficient\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}}{x}^n=1.\frac{1}{n!}+1.\frac{1}{(n-1)!}+\cdots +1.1$

$=1.1+1.\frac{1}{2!}+1.\frac{1}{3!}+\cdots +\frac{1}{n!}$

$=-1+\left(1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\cdots +\frac{1}{n!}\right)$

$\begin{array}{l}=-1+e\\ =e-1\end{array}$

$B_{n-1}=\text{coefficient\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}}{x}^{n-1}=1.1+1.\frac{1}{2!}+\cdots +1.\frac{1}{(n-1)!}$

$=-1+\left[1+\frac{{\displaystyle 1}}{{\displaystyle 1!}}+\frac{{\displaystyle 1}}{{\displaystyle 2!}}+\cdots +\frac{{\displaystyle 1}}{{\displaystyle n!}}\right]-\frac{1}{n!}$

$\begin{array}{l}{B}_{n-1}=e-1-\frac{1}{n!}\\ B_n-B_{n-1}=(e-1)-\left(e-1-\frac{1}{n!}\right)\\ =\frac{1}{n!}\end{array}$