A) \[\frac{1}{n!}-\frac{1}{(n-1)!}\]
B) \[\frac{1}{n!}\]
C) \[\frac{1}{(n-1)!}\]
D) \[\frac{1}{n!}+\frac{1}{(n-1)!}\]
Correct Answer: B
Solution :
[b] We have \[(1-x)({{B}_{0}}+{{B}_{1}}x+{{B}_{2}}{{x}^{2}}+....)\] \[.....+{{B}_{n-1}}{{x}^{n-1}}+{{B}_{n}}{{x}^{n}}+...)\] \[={{e}^{x}}=1+x+\frac{{{x}^{2}}}{2!}+\frac{{{x}^{3}}}{3!}+....+\frac{{{x}^{n}}}{n!}+.......(1)\] Hence equating the coefficients of \[{{x}^{n}}\] on both sides of (1) we get \[{{B}_{n}}-{{B}_{n-1}}=\frac{1}{n!}.\]
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