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k-1-i-0-k-1-n-i-k-is-this-really-mean-something-




Question Number 37859 by kunal1234523 last updated on 18/Jun/18
Σ_(k=1) ^∞ ((Π_(i=0) ^(k−1) (n−i))/(k!))  is this really mean something
$$\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\underset{{i}=\mathrm{0}} {\overset{{k}−\mathrm{1}} {\prod}}\left({n}−{i}\right)}{{k}!} \\ $$$$\mathrm{is}\:\mathrm{this}\:\mathrm{really}\:\mathrm{mean}\:\mathrm{something} \\ $$
Commented by math khazana by abdo last updated on 18/Jun/18
Π_(i=0) ^(k−1) (n−i)=n(n−1)(n−2)...(n−k+1)  =((n(n−1)....(n−k+1)(n−k)(n−k−1....2.1)/((n−k)!))  =((n!)/((n−k)!)) ⇒((Π_(i=0) ^(k−1) (n−i))/(k!)) = C_n ^(k )  ⇒  Σ_(k=1) ^∞   (...) =Σ_(k=1) ^∞   C_n ^k    and this is a nonsens     because k?must be ≤n by definition of   combination...
$$\prod_{{i}=\mathrm{0}} ^{{k}−\mathrm{1}} \left({n}−{i}\right)={n}\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)…\left({n}−{k}+\mathrm{1}\right) \\ $$$$=\frac{{n}\left({n}−\mathrm{1}\right)….\left({n}−{k}+\mathrm{1}\right)\left({n}−{k}\right)\left({n}−{k}−\mathrm{1}….\mathrm{2}.\mathrm{1}\right.}{\left({n}−{k}\right)!} \\ $$$$=\frac{{n}!}{\left({n}−{k}\right)!}\:\Rightarrow\frac{\prod_{{i}=\mathrm{0}} ^{{k}−\mathrm{1}} \left({n}−{i}\right)}{{k}!}\:=\:{C}_{{n}} ^{{k}\:} \:\Rightarrow \\ $$$$\sum_{{k}=\mathrm{1}} ^{\infty} \:\:\left(…\right)\:=\sum_{{k}=\mathrm{1}} ^{\infty} \:\:{C}_{{n}} ^{{k}} \:\:\:{and}\:{this}\:{is}\:{a}\:{nonsens}\:\:\: \\ $$$${because}\:{k}?{must}\:{be}\:\leqslant{n}\:{by}\:{definition}\:{of}\: \\ $$$${combination}… \\ $$

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