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Question Number 66795 by mathmax by abdo last updated on 19/Aug/19
let f(x) =e^(−x) ln(1+x^2 ) 1) calculate f^((n)) (0) 2) developp f at integr serie
$${let}\:{f}\left({x}\right)\:={e}^{−{x}} {ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right) \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$
Commented by mathmax by abdo last updated on 26/Aug/19
1) we have f(x) =e^(−x) ln(1+x^2 ) ⇒ f^((n)) (x) =Σ_(k=0) ^n C_n ^k (ln(1+x^2 ))^((k)) (e^(−x) )^((n−k)) =(−1)^n e^(−x) ln(1+x^2 ) +Σ_(k=1) ^n C_n ^k {ln(1+x^2 )}^((k)) (e^(−x) )^((n−k)) (ln(1+x^2 ))^((1)) =((2x)/(1+x^2 )) ⇒{ln(1+x^2 )}^((k)) =(((2x)/(1+x^2 )))^((k−1)) =(((2x)/((x−i)(x+i))))^((k−1)) ={(1/(x+i))+(1/(x−i))}^((k−1)) =(((−1)^(k−1) (k−1)!)/((x+i)^k )) +(((−1)^(k−1) (k−1)!)/((x−i)^k )) =(−1)^(k−1) (k−1)!{(1/((x+i)^k ))+(1/((x−i)^k ))} =(−1)^(k−1) (k−1)!×((2Re((x+i)^k ))/((x^2 +1)^k )) ⇒ f^((n)) (x) =(−1)^n e^(−x) ln(1+x^2 ) +Σ_(k=1) ^n C_n ^k ((2(−1)^(k−1) (k−1)! Re((x+i)^k ))/((x^2 +1)^k ))×(−1)^(n−k) e^(−x) ⇒f^((n)) (0) =Σ_(k=1) ^n C_n ^k 2(−1)^(k−1) (k−1)!cos(((kπ)/2))(−1)^(n−k) =2Σ_(k=1) ^n (−1)^(n−1) C_n ^k cos(((kπ)/2))
$$\left.\mathrm{1}\right)\:{we}\:{have}\:{f}\left({x}\right)\:={e}^{−{x}} {ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\:\Rightarrow \\ $$$${f}^{\left({n}\right)} \left({x}\right)\:=\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \left({ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\right)^{\left({k}\right)} \:\left({e}^{−{x}} \right)^{\left({n}−{k}\right)} \\ $$$$=\left(−\mathrm{1}\right)^{{n}} \:{e}^{−{x}} {ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\:+\sum_{{k}=\mathrm{1}} ^{{n}} \:{C}_{{n}} ^{{k}} \left\{{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\right\}^{\left({k}\right)} \left({e}^{−{x}} \right)^{\left({n}−{k}\right)} \\ $$$$\left({ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\right)^{\left(\mathrm{1}\right)} \:=\frac{\mathrm{2}{x}}{\mathrm{1}+{x}^{\mathrm{2}} }\:\Rightarrow\left\{{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\right\}^{\left({k}\right)} \:=\left(\frac{\mathrm{2}{x}}{\mathrm{1}+{x}^{\mathrm{2}} }\right)^{\left({k}−\mathrm{1}\right)} \\ $$$$=\left(\frac{\mathrm{2}{x}}{\left({x}−{i}\right)\left({x}+{i}\right)}\right)^{\left({k}−\mathrm{1}\right)} =\left\{\frac{\mathrm{1}}{{x}+{i}}+\frac{\mathrm{1}}{{x}−{i}}\right\}^{\left({k}−\mathrm{1}\right)} =\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \left({k}−\mathrm{1}\right)!}{\left({x}+{i}\right)^{{k}} } \\ $$$$+\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \left({k}−\mathrm{1}\right)!}{\left({x}−{i}\right)^{{k}} }\:=\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \left({k}−\mathrm{1}\right)!\left\{\frac{\mathrm{1}}{\left({x}+{i}\right)^{{k}} }+\frac{\mathrm{1}}{\left({x}−{i}\right)^{{k}} }\right\} \\ $$$$=\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \left({k}−\mathrm{1}\right)!×\frac{\mathrm{2}{Re}\left(\left({x}+{i}\right)^{{k}} \right)}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)^{{k}} }\:\Rightarrow \\ $$$${f}^{\left({n}\right)} \left({x}\right)\:=\left(−\mathrm{1}\right)^{{n}} \:{e}^{−{x}} {ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right) \\ $$$$+\sum_{{k}=\mathrm{1}} ^{{n}} \:{C}_{{n}} ^{{k}} \:\:\:\frac{\mathrm{2}\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \left({k}−\mathrm{1}\right)!\:{Re}\left(\left({x}+{i}\right)^{{k}} \right)}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)^{{k}} }×\left(−\mathrm{1}\right)^{{n}−{k}} \:{e}^{−{x}} \\ $$$$\Rightarrow{f}^{\left({n}\right)} \left(\mathrm{0}\right)\:=\sum_{{k}=\mathrm{1}} ^{{n}} \:{C}_{{n}} ^{{k}} \:\mathrm{2}\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \left({k}−\mathrm{1}\right)!{cos}\left(\frac{{k}\pi}{\mathrm{2}}\right)\left(−\mathrm{1}\right)^{{n}−{k}} \\ $$$$=\mathrm{2}\sum_{{k}=\mathrm{1}} ^{{n}} \:\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \:{C}_{{n}} ^{{k}} \:\:{cos}\left(\frac{{k}\pi}{\mathrm{2}}\right) \\ $$$$ \\ $$$$ \\ $$
Commented by mathmax by abdo last updated on 26/Aug/19
f^((n)) (0) =2(−1)^(n−1) Σ_(k=1) ^n (k−1)! C_n ^k cos(((kπ)/2))
$${f}^{\left({n}\right)} \left(\mathrm{0}\right)\:=\mathrm{2}\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \:\sum_{{k}=\mathrm{1}} ^{{n}} \left({k}−\mathrm{1}\right)!\:{C}_{{n}} ^{{k}} \:{cos}\left(\frac{{k}\pi}{\mathrm{2}}\right) \\ $$
Commented by mathmax by abdo last updated on 26/Aug/19
2) we have f(x) =Σ_(n=0) ^∞ ((f^((n)) (0))/(n!)) x^n =f(o) +Σ_(n=1) ^∞ ((f^((n)) (0))/(n!))x^n f(x) =2Σ_(n=1) ^∞ (((−1)^(n−1) )/(n!))( Σ_(k=1) ^n (k−1)! C_n ^k cos(((kπ)/2)))x^n =2 Σ_(n=1) ^∞ (−1)^(n−1) (Σ_(k=1) ^n (k−1)!×(1/(k!(n−k)!)))x^n =2 Σ_(n=1) ^∞ (−1)^(n−1) (Σ_(k=1) ^n (1/(k(n−k)!)))x^n
$$\left.\mathrm{2}\right)\:{we}\:{have}\:{f}\left({x}\right)\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{f}^{\left({n}\right)} \left(\mathrm{0}\right)}{{n}!}\:{x}^{{n}} \:={f}\left({o}\right)\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{{f}^{\left({n}\right)} \left(\mathrm{0}\right)}{{n}!}{x}^{{n}} \\ $$$${f}\left({x}\right)\:=\mathrm{2}\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{n}!}\left(\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\left({k}−\mathrm{1}\right)!\:{C}_{{n}} ^{{k}} \:{cos}\left(\frac{{k}\pi}{\mathrm{2}}\right)\right){x}^{{n}} \\ $$$$=\mathrm{2}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left(\sum_{{k}=\mathrm{1}} ^{{n}} \left({k}−\mathrm{1}\right)!×\frac{\mathrm{1}}{{k}!\left({n}−{k}\right)!}\right){x}^{{n}} \\ $$$$=\mathrm{2}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left(\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}\left({n}−{k}\right)!}\right){x}^{{n}} \\ $$

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