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Question Number 40105 by maxmathsup by imad last updated on 15/Jul/18
let f(x) = x^n e^(−2nx) with n integr natural calculate f^((n)) (0).
$${let}\:\:{f}\left({x}\right)\:=\:{x}^{{n}} \:{e}^{−\mathrm{2}{nx}} \:\:\:\:{with}\:{n}\:{integr}\:{natural} \\ $$$$\:{calculate}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right). \\ $$
Commented by maxmathsup by imad last updated on 17/Jul/18
leibniz formula give f^((p)) (x)=Σ_(k=0) ^p C_p ^k (x^n )^((k)) (e^(−2nx) )^((p−k)) but (x^n )^((k)) = n(n−1)....(n−k+1) x^(n−k) if k≤n and (x^n )^((k)) =0 if k>n ⇒ (x^n )^((k)) =((n!)/((n−k)!)) x^(n−k) also we have {e^(−2n) }^((p−k)) =(−2n)^(p−k) e^(−2nx) ⇒ f^((p)) (x)=Σ_(k=0) ^p C_p ^k ((n!)/((n−k)!)) x^(n−k) (−2n)^(p−k) e^(−2nx) let suppose n≤p f^((p)) (x)= Σ_(k=0) ^(n−1) C_p ^k ((n!)/((n−k)!)) x^(n−k) (−2n)^(p−k) e^(−2nx ) + C_p ^n ((n!)/(0!))(−2n)^(p−n) e^(−2nx) +Σ_(k=n+1) ^p C_p ^k ((n!)/((n−k)!)) x^(n−k) (−2n)^(p−k) e^(−2nx) ⇒ f^((p)) (0) = C_p ^n n! (−2n)^(p−n) ⇒ f^((n)) (0) =n! C_n ^n =n!
$${leibniz}\:{formula}\:{give}\:\:{f}^{\left({p}\right)} \left({x}\right)=\sum_{{k}=\mathrm{0}} ^{{p}} \:\:{C}_{{p}} ^{{k}} \:\:\left({x}^{{n}} \right)^{\left({k}\right)} \:\left({e}^{−\mathrm{2}{nx}} \right)^{\left({p}−{k}\right)} \:\:{but}\: \\ $$$$\left({x}^{{n}} \right)^{\left({k}\right)} \:=\:{n}\left({n}−\mathrm{1}\right)….\left({n}−{k}+\mathrm{1}\right)\:{x}^{{n}−{k}} \:\:{if}\:{k}\leqslant{n}\:{and}\:\left({x}^{{n}} \right)^{\left({k}\right)} =\mathrm{0}\:{if}\:{k}>{n}\:\Rightarrow \\ $$$$\left({x}^{{n}} \right)^{\left({k}\right)} \:\:=\frac{{n}!}{\left({n}−{k}\right)!}\:{x}^{{n}−{k}} \:\:\:{also}\:{we}\:{have}\:\left\{{e}^{−\mathrm{2}{n}} \right\}^{\left({p}−{k}\right)} =\left(−\mathrm{2}{n}\right)^{{p}−{k}} \:{e}^{−\mathrm{2}{nx}} \:\Rightarrow \\ $$$${f}^{\left({p}\right)} \left({x}\right)=\sum_{{k}=\mathrm{0}} ^{{p}} \:\:{C}_{{p}} ^{{k}} \:\:\:\:\:\frac{{n}!}{\left({n}−{k}\right)!}\:{x}^{{n}−{k}} \:\:\left(−\mathrm{2}{n}\right)^{{p}−{k}} \:{e}^{−\mathrm{2}{nx}} \:\:\:{let}\:{suppose}\:{n}\leqslant{p} \\ $$$${f}^{\left({p}\right)} \left({x}\right)=\:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:{C}_{{p}} ^{{k}} \:\:\frac{{n}!}{\left({n}−{k}\right)!}\:{x}^{{n}−{k}} \left(−\mathrm{2}{n}\right)^{{p}−{k}} \:{e}^{−\mathrm{2}{nx}\:} \:+\:{C}_{{p}} ^{{n}} \:\:\:\frac{{n}!}{\mathrm{0}!}\left(−\mathrm{2}{n}\right)^{{p}−{n}} \:{e}^{−\mathrm{2}{nx}} \\ $$$$+\sum_{{k}={n}+\mathrm{1}} ^{{p}} \:{C}_{{p}} ^{{k}} \:\:\:\:\frac{{n}!}{\left({n}−{k}\right)!}\:{x}^{{n}−{k}} \left(−\mathrm{2}{n}\right)^{{p}−{k}} \:{e}^{−\mathrm{2}{nx}} \:\Rightarrow \\ $$$${f}^{\left({p}\right)} \left(\mathrm{0}\right)\:=\:{C}_{{p}} ^{{n}} {n}!\:\left(−\mathrm{2}{n}\right)^{{p}−{n}} \:\Rightarrow\:{f}^{\left({n}\right)} \left(\mathrm{0}\right)\:={n}!\:{C}_{{n}} ^{{n}} \:\:={n}! \\ $$

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