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Question Number 42999 by abdo.msup.com last updated on 06/Sep/18
let f(x)=(√x)+(1/(x−1)) 1) calculate f^((n)) (2) 2) if f(x) =Σ_(n=0) ^∞ a_n (x−2)^n find the sequence a_n
$${let}\:{f}\left({x}\right)=\sqrt{{x}}+\frac{\mathrm{1}}{{x}−\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left(\mathrm{2}\right) \\ $$$$\left.\mathrm{2}\right)\:{if}\:{f}\left({x}\right)\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:{a}_{{n}} \left({x}−\mathrm{2}\right)^{{n}} \:{find}\:{the} \\ $$$${sequence}\:{a}_{{n}} \\ $$
Commented by maxmathsup by imad last updated on 08/Sep/18
1) first let find f^((n)) (x) we have f^((n)) (x)=((√x))^n +((1/(x−1)))^n but {(1/(x−1))}^((n)) = (((−1)^n n!)/((x−1)^(n+1) )) let determine ((√x))^n ={x^(1/2) }^((n)) let α fromQ {x^α }^((1)) =α x^(α−1) ⇒{x^α }^((2)) =α(α−1)x^(α−2) ⇒{x^α }^((n)) =α(α−1)...(α−n+1)x^(α−n) so { x^(1/2) }^((n)) =(1/2)((1/2)−1)((1/2)−2)....((1/2) −n+1)x^((1/2)−n) =(1/2)(−(1/2))(−(3/2))....(((1−2n+2))/2))x^((1/2)−n) =(1/2)(−(1/2))(−(3/2))....(−((2n−3)/2)) x^((1/2)−n) ⇒ f^((n)) (x)=(1/2)(−(1/2))(−(3/2))=(−((2n−3)/2))x^((1/2)−n) + (((−1)^n n!)/((x−1)^(n+1) )) .⇒ f^((n)) (2) =(1/2)(−(1/2))(−(3/2)).....(−((2n−3)/2))2^((1/2)−n) +(−1)^n n! = (1/(2^n (√2))) (−(1/2))(−(3/2))...(−((2n−3)/2)) +(−1)^n n! . 2) developpement of f(x) at integr serie seris at v(2) give f(x) =Σ_(n=0) ^∞ ((f^((n)) (2))/(n!)) (x−2)^n ⇒a_n =((f^((n)) (2))/(n!)) ⇒ a_n = (1/((n!)2^n (√2)))(−(1/2))(−(3/2))...(−((2n−3)/2)) +(−1)^n . nt
$$\left.\mathrm{1}\right)\:{first}\:{let}\:{find}\:{f}^{\left({n}\right)} \left({x}\right)\:\:{we}\:{have}\:{f}^{\left({n}\right)} \left({x}\right)=\left(\sqrt{{x}}\right)^{{n}} \:+\left(\frac{\mathrm{1}}{{x}−\mathrm{1}}\right)^{{n}} \:\:{but} \\ $$$$\left\{\frac{\mathrm{1}}{{x}−\mathrm{1}}\right\}^{\left({n}\right)} \:=\:\frac{\left(−\mathrm{1}\right)^{{n}} {n}!}{\left({x}−\mathrm{1}\right)^{{n}+\mathrm{1}} }\:{let}\:{determine}\:\left(\sqrt{{x}}\right)^{{n}} \:=\left\{{x}^{\frac{\mathrm{1}}{\mathrm{2}}} \right\}^{\left({n}\right)} {let}\:\alpha\:{fromQ} \\ $$$$\left\{{x}^{\alpha} \right\}^{\left(\mathrm{1}\right)} \:=\alpha\:{x}^{\alpha−\mathrm{1}} \:\Rightarrow\left\{{x}^{\alpha} \right\}^{\left(\mathrm{2}\right)} =\alpha\left(\alpha−\mathrm{1}\right){x}^{\alpha−\mathrm{2}} \:\Rightarrow\left\{{x}^{\alpha} \right\}^{\left({n}\right)} \:=\alpha\left(\alpha−\mathrm{1}\right)…\left(\alpha−{n}+\mathrm{1}\right){x}^{\alpha−{n}} \\ $$$${so}\:\left\{\:{x}^{\frac{\mathrm{1}}{\mathrm{2}}} \right\}^{\left({n}\right)} =\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\mathrm{1}}{\mathrm{2}}−\mathrm{1}\right)\left(\frac{\mathrm{1}}{\mathrm{2}}−\mathrm{2}\right)….\left(\frac{\mathrm{1}}{\mathrm{2}}\:−{n}+\mathrm{1}\right){x}^{\frac{\mathrm{1}}{\mathrm{2}}−{n}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left(−\frac{\mathrm{1}}{\mathrm{2}}\right)\left(−\frac{\mathrm{3}}{\mathrm{2}}\right)….\left(\frac{\left.\mathrm{1}−\mathrm{2}{n}+\mathrm{2}\right)}{\mathrm{2}}\right){x}^{\frac{\mathrm{1}}{\mathrm{2}}−{n}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left(−\frac{\mathrm{1}}{\mathrm{2}}\right)\left(−\frac{\mathrm{3}}{\mathrm{2}}\right)….\left(−\frac{\mathrm{2}{n}−\mathrm{3}}{\mathrm{2}}\right)\:{x}^{\frac{\mathrm{1}}{\mathrm{2}}−{n}} \:\Rightarrow \\ $$$${f}^{\left({n}\right)} \left({x}\right)=\frac{\mathrm{1}}{\mathrm{2}}\left(−\frac{\mathrm{1}}{\mathrm{2}}\right)\left(−\frac{\mathrm{3}}{\mathrm{2}}\right)=\left(−\frac{\mathrm{2}{n}−\mathrm{3}}{\mathrm{2}}\right){x}^{\frac{\mathrm{1}}{\mathrm{2}}−{n}} \:+\:\frac{\left(−\mathrm{1}\right)^{{n}} {n}!}{\left({x}−\mathrm{1}\right)^{{n}+\mathrm{1}} }\:.\Rightarrow \\ $$$${f}^{\left({n}\right)} \left(\mathrm{2}\right)\:=\frac{\mathrm{1}}{\mathrm{2}}\left(−\frac{\mathrm{1}}{\mathrm{2}}\right)\left(−\frac{\mathrm{3}}{\mathrm{2}}\right)…..\left(−\frac{\mathrm{2}{n}−\mathrm{3}}{\mathrm{2}}\right)\mathrm{2}^{\frac{\mathrm{1}}{\mathrm{2}}−{n}} \:\:+\left(−\mathrm{1}\right)^{{n}} {n}! \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{2}^{{n}} \sqrt{\mathrm{2}}}\:\left(−\frac{\mathrm{1}}{\mathrm{2}}\right)\left(−\frac{\mathrm{3}}{\mathrm{2}}\right)…\left(−\frac{\mathrm{2}{n}−\mathrm{3}}{\mathrm{2}}\right)\:+\left(−\mathrm{1}\right)^{{n}} {n}!\:. \\ $$$$\left.\mathrm{2}\right)\:{developpement}\:{of}\:{f}\left({x}\right)\:{at}\:{integr}\:{serie}\:{seris}\:{at}\:{v}\left(\mathrm{2}\right)\:{give} \\ $$$${f}\left({x}\right)\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{{f}^{\left({n}\right)} \left(\mathrm{2}\right)}{{n}!}\:\left({x}−\mathrm{2}\right)^{{n}} \:\Rightarrow{a}_{{n}} =\frac{{f}^{\left({n}\right)} \left(\mathrm{2}\right)}{{n}!}\:\Rightarrow \\ $$$${a}_{{n}} =\:\:\frac{\mathrm{1}}{\left({n}!\right)\mathrm{2}^{{n}} \sqrt{\mathrm{2}}}\left(−\frac{\mathrm{1}}{\mathrm{2}}\right)\left(−\frac{\mathrm{3}}{\mathrm{2}}\right)…\left(−\frac{\mathrm{2}{n}−\mathrm{3}}{\mathrm{2}}\right)\:+\left(−\mathrm{1}\right)^{{n}} \:. \\ $$$$ \\ $$$${nt} \\ $$

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