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If $n$ be a positive integer such that $n \ge 3$, then the value of the sum to $n$ terms of the series $1 . n - \frac{{\left( {n\, - \,1} \right)}}{{1\,\,!}} (n - 1) + \frac{{\left( {n\, - \,1} \right)\,\,\left( {n\, - \,2} \right)}}{{2\,\,!}} (n - 2) $$- \frac{{\left( {n\, - \,1} \right)\,\,\left( {n\, - \,2} \right)\,\,\left( {n\, - \,3} \right)}}{{3\,\,!}} (n - 3) + ......$ is
$0$
$1$
$-1$
none of these
Solution
$n [ {{\,^{n\, – \,1}}{C_0}\,\, – \,{\,^{n\, – \,1}}{C_1}\,\,.\,\,\left( {1\,\, – \,\,\frac{1}{n}} \right)\,\, + \,{\,^{n\, – \,1}}{C_2}}\,\,.$ ${\,\,\left( {1\,\, – \,\,\frac{2}{n}} \right)\,\, – \,{\,^{n\, – \,1}}{C^3}\,\,.\,\,\left( {1\,\, – \,\,\frac{3}{n}} \right)\,\, + \,\,……\,} ]$
$= n \left[ {\,\underbrace {^{n\, – \,1}{C_0}\,\, – \,{\,^{n\, – \,1}}{C_1}\,\, + \,{\,^{n\, – \,1}}{C_2}\,\, – \,{\,^{n\, – \,1}}{C_3}}_{Vanishes}\,\, + \,\,……} \right]$ $+$ $\left[ {{\,^{n\, – \,1}}{C_1}\,\, – \,\,2\,\,.\,{\,^{n\, – \,1}}{C_2}\,\, + \,\,.\,\,.\,{\,^{n\, – \,1}}{C_3}\,\, – \,\,4\,\,.\,{\,^{n\, – \,1}}{C_3}\,\, + \,\,……} \right]$
For $2^{nd}$ consider expansion of $(1 + x)^{n – 1}$ and differentiate $w.r.t. x$ and put $x = – 1$
