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let-f-x-cosx-x-2-1-1-calculate-f-n-x-then-f-n-0-2-calculste-f-n-0-3-developp-f-at-integr-serie-

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Question Number 56583 by maxmathsup by imad last updated on 18/Mar/19
let f(x) =((cosx)/(x^2 +1)) 1) calculate f^((n)) (x) then f^((n)) (0) 2)calculste f^((n)) (0) 3) developp f at integr serie .
$${let}\:{f}\left({x}\right)\:=\frac{{cosx}}{{x}^{\mathrm{2}} \:+\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left({x}\right)\:\:{then}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right){calculste}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{3}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie}\:. \\ $$
Commented by maxmathsup by imad last updated on 20/Mar/19
1) we have f(x) =((cosx)/((x−i)(x+i))) =(1/(2i)){((cosx)/(x−i)) −((cosx)/(x+i))} ⇒ f^((n)) (x)=(1/(2i)){ (((cosx)/(x−i)))^((n)) −(((cosx)/(x+i)))^((n)) } leibniz formula give (((cosx)/(x+i)))^((n)) =Σ_(k=0) ^n C_n ^k ((1/(x+i)))^((k)) (cosx)^((n−k)) =Σ_(k=0) ^n C_n ^k (((−1)^k k!)/((x+i)^(k+1) )) cos(x +(n−k)(π/2)) also for the same raison (((cosx)/(x−i)))^((n)) =Σ_(k=0) ^n C_n ^k (((−1)^k k!)/((x−i)^(k+1) )) cos(x+(n−k)(π/2)) ⇒f^((n)) (x) =(1/(2i)) Σ_(k=0) ^n C_n ^k (−1)^k k!cos(x+(n−k)(π/2)){(1/((x−i)^(k+1) )) −(1/((x+i)^(k+1) ))} f^((n)) (x)=(1/(2i)) Σ_(k=0) ^n C_n ^k (−1)^k k! cos(x+(n−k)(π/2))(((x+i)^(k+1) −(x−i)^(k+1) )/((x^2 +1)^(k+1) )) 2)x=0 ⇒f^((n)) (0) =(1/(2i)) Σ_(k=0) ^n C_n ^k (−1)^k k! cos((n−k)(π/2)) ((2i Im(i^(k+1) ))/1) ⇒ f^((n)) (0) =Σ_(k=0) ^n C_n ^k (−1)^k k! cos((n−k)(π/2))sin((k+1)(π/2)) . 3) f(x) =Σ_(n=0) ^∞ ((f^((n)) (0))/(n!)) x^n =Σ_(n=0) ^∞ (1/(n!)){Σ_(k=0) ^n ((n!)/(k!(n−k)!))(−1)^k k! cos((n−k)(π/2))sin((k+1)(π/2)}x^n ⇒ f(x) =Σ_(n=0) ^∞ {Σ_(k=0) ^n (((−1)^k )/((n−k)!)) cos((n−k)(π/2))sin((k+1)(π/2))}x^n .
$$\left.\mathrm{1}\right)\:{we}\:{have}\:{f}\left({x}\right)\:=\frac{{cosx}}{\left({x}−{i}\right)\left({x}+{i}\right)}\:=\frac{\mathrm{1}}{\mathrm{2}{i}}\left\{\frac{{cosx}}{{x}−{i}}\:−\frac{{cosx}}{{x}+{i}}\right\}\:\Rightarrow \\ $$$${f}^{\left({n}\right)} \left({x}\right)=\frac{\mathrm{1}}{\mathrm{2}{i}}\left\{\:\:\left(\frac{{cosx}}{{x}−{i}}\right)^{\left({n}\right)} −\left(\frac{{cosx}}{{x}+{i}}\right)^{\left({n}\right)} \right\}\:{leibniz}\:{formula}\:{give} \\ $$$$\left(\frac{{cosx}}{{x}+{i}}\right)^{\left({n}\right)} \:=\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \:\:\left(\frac{\mathrm{1}}{{x}+{i}}\right)^{\left({k}\right)} \left({cosx}\right)^{\left({n}−{k}\right)} \:=\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \:\frac{\left(−\mathrm{1}\right)^{{k}} {k}!}{\left({x}+{i}\right)^{{k}+\mathrm{1}} }\:{cos}\left({x}\:+\left({n}−{k}\right)\frac{\pi}{\mathrm{2}}\right) \\ $$$${also}\:{for}\:{the}\:{same}\:{raison}\:\:\left(\frac{{cosx}}{{x}−{i}}\right)^{\left({n}\right)} \:\:=\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \:\:\frac{\left(−\mathrm{1}\right)^{{k}} {k}!}{\left({x}−{i}\right)^{{k}+\mathrm{1}} }\:{cos}\left({x}+\left({n}−{k}\right)\frac{\pi}{\mathrm{2}}\right) \\ $$$$\Rightarrow{f}^{\left({n}\right)} \left({x}\right)\:=\frac{\mathrm{1}}{\mathrm{2}{i}}\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\:{C}_{{n}} ^{{k}} \:\left(−\mathrm{1}\right)^{{k}} {k}!{cos}\left({x}+\left({n}−{k}\right)\frac{\pi}{\mathrm{2}}\right)\left\{\frac{\mathrm{1}}{\left({x}−{i}\right)^{{k}+\mathrm{1}} }\:−\frac{\mathrm{1}}{\left({x}+{i}\right)^{{k}+\mathrm{1}} }\right\} \\ $$$${f}^{\left({n}\right)} \left({x}\right)=\frac{\mathrm{1}}{\mathrm{2}{i}}\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \left(−\mathrm{1}\right)^{{k}} {k}!\:{cos}\left({x}+\left({n}−{k}\right)\frac{\pi}{\mathrm{2}}\right)\frac{\left({x}+{i}\right)^{{k}+\mathrm{1}} −\left({x}−{i}\right)^{{k}+\mathrm{1}} }{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{{k}+\mathrm{1}} } \\ $$$$\left.\mathrm{2}\right){x}=\mathrm{0}\:\Rightarrow{f}^{\left({n}\right)} \left(\mathrm{0}\right)\:=\frac{\mathrm{1}}{\mathrm{2}{i}}\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \left(−\mathrm{1}\right)^{{k}} {k}!\:{cos}\left(\left({n}−{k}\right)\frac{\pi}{\mathrm{2}}\right)\:\frac{\mathrm{2}{i}\:{Im}\left({i}^{{k}+\mathrm{1}} \right)}{\mathrm{1}}\:\Rightarrow \\ $$$${f}^{\left({n}\right)} \left(\mathrm{0}\right)\:=\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \left(−\mathrm{1}\right)^{{k}} {k}!\:{cos}\left(\left({n}−{k}\right)\frac{\pi}{\mathrm{2}}\right){sin}\left(\left({k}+\mathrm{1}\right)\frac{\pi}{\mathrm{2}}\right)\:. \\ $$$$\left.\mathrm{3}\right)\:{f}\left({x}\right)\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{{f}^{\left({n}\right)} \left(\mathrm{0}\right)}{{n}!}\:{x}^{{n}} \\ $$$$=\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}}{{n}!}\left\{\sum_{{k}=\mathrm{0}} ^{{n}} \:\frac{{n}!}{{k}!\left({n}−{k}\right)!}\left(−\mathrm{1}\right)^{{k}} {k}!\:{cos}\left(\left({n}−{k}\right)\frac{\pi}{\mathrm{2}}\right){sin}\left(\left({k}+\mathrm{1}\right)\frac{\pi}{\mathrm{2}}\right\}{x}^{{n}} \:\Rightarrow\right. \\ $$$${f}\left({x}\right)\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\left\{\sum_{{k}=\mathrm{0}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{\left({n}−{k}\right)!}\:{cos}\left(\left({n}−{k}\right)\frac{\pi}{\mathrm{2}}\right){sin}\left(\left({k}+\mathrm{1}\right)\frac{\pi}{\mathrm{2}}\right)\right\}{x}^{{n}} \:. \\ $$

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