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Question Number 68239 by mathmax by abdo last updated on 07/Sep/19
let f(x) =e^(−2x) ln(1+x^2 ) developp f at integr serie.
$${let}\:{f}\left({x}\right)\:={e}^{−\mathrm{2}{x}} {ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right) \\ $$$${developp}\:{f}\:{at}\:{integr}\:{serie}. \\ $$
Commented by mathmax by abdo last updated on 09/Sep/19
first let determine f^((n)) (x) we have f^((n)) (x)=Σ_(k=0) ^n C_n ^k (ln(1+x^2 ))^((k)) (e^(−2x) )^((n−k)) =ln(1+x^2 )(−2)^n e^(−2x) +Σ_(k=1) ^n C_n ^k (ln(1+x^2 ))^((k)) (−2)^(n−k) e^(−2x) let find (ln(1+x^2 ))^((k)) for k≥1 let w(x)=ln(1+x^2 ) we have w^′ (x) =((2x)/(1+x^2 )) =((2x)/((x−i)(x+i))) =(1/(x+i))+(1/(x−i)) ⇒ w^((k)) (x) =((1/(x+i)))^((k−1)) +((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)!{(((x+i)^k +(x−i)^k )/((x^2 +1)^k ))}=(((−1)^(k−1) (k−1)!2Re(x+i)^k )/((x^2 +1)^k )) we have x+i =(√(1+x^2 ))e^(iarctan((1/x))) ⇒(x+i)^k =(1+x^2 )^(k/2) e^(ikarctan((1/x))) w^((k)) (x) =2(((−1)^(k−1) (k−1)!(1+x^2 )^(k/2) cos(k arctan((1/x))))/((x^2 +1)^k )) ⇒ f^((n)) (x)=(−2)^n e^(−2x) ln(1+x^2 ) +Σ_(k=1) ^n C_n ^k ((2(−1)^(k−1) (k−1)!(1+x^2 )^(k/2) cos(karctan((1/x))))/((x^2 +1)^k ))(−2)^(n−k) e^(−2x)
$${first}\:{let}\:{determine}\:{f}^{\left({n}\right)} \left({x}\right)\:\:\:{we}\:{have} \\ $$$${f}^{\left({n}\right)} \left({x}\right)=\sum_{{k}=\mathrm{0}} ^{{n}} {C}_{{n}} ^{{k}} \:\left({ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\right)^{\left({k}\right)} \left({e}^{−\mathrm{2}{x}} \right)^{\left({n}−{k}\right)} \\ $$$$={ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(−\mathrm{2}\right)^{{n}} {e}^{−\mathrm{2}{x}} \:+\sum_{{k}=\mathrm{1}} ^{{n}} \:{C}_{{n}} ^{{k}} \left({ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\right)^{\left({k}\right)} \left(−\mathrm{2}\right)^{{n}−{k}} {e}^{−\mathrm{2}{x}} \\ $$$${let}\:\:{find}\:\left({ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\right)^{\left({k}\right)} \:{for}\:{k}\geqslant\mathrm{1}\:\:{let}\:{w}\left({x}\right)={ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\:{we}\:{have} \\ $$$${w}^{'} \left({x}\right)\:=\frac{\mathrm{2}{x}}{\mathrm{1}+{x}^{\mathrm{2}} }\:=\frac{\mathrm{2}{x}}{\left({x}−{i}\right)\left({x}+{i}\right)}\:=\frac{\mathrm{1}}{{x}+{i}}+\frac{\mathrm{1}}{{x}−{i}}\:\Rightarrow \\ $$$${w}^{\left({k}\right)} \left({x}\right)\:=\left(\frac{\mathrm{1}}{{x}+{i}}\right)^{\left({k}−\mathrm{1}\right)} \:+\left(\frac{\mathrm{1}}{{x}−{i}}\right)^{\left({k}−\mathrm{1}\right)} \:=\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \left({k}−\mathrm{1}\right)!}{\left({x}+{i}\right)^{{k}} }+\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \left({k}−\mathrm{1}\right)}{\left({x}−{i}\right)^{{k}} } \\ $$$$=\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \left({k}−\mathrm{1}\right)!\left\{\:\frac{\mathrm{1}}{\left({x}+{i}\right)^{{k}} }+\frac{\mathrm{1}}{\left({x}−{i}\right)^{{k}} }\right\} \\ $$$$=\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \left({k}−\mathrm{1}\right)!\left\{\frac{\left({x}+{i}\right)^{{k}} +\left({x}−{i}\right)^{{k}} }{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)^{{k}} }\right\}=\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \left({k}−\mathrm{1}\right)!\mathrm{2}{Re}\left({x}+{i}\right)^{{k}} }{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)^{{k}} } \\ $$$${we}\:{have}\:{x}+{i}\:=\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }{e}^{{iarctan}\left(\frac{\mathrm{1}}{{x}}\right)} \:\Rightarrow\left({x}+{i}\right)^{{k}} \:=\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\frac{{k}}{\mathrm{2}}} \:{e}^{{ikarctan}\left(\frac{\mathrm{1}}{{x}}\right)} \\ $$$${w}^{\left({k}\right)} \left({x}\right)\:=\mathrm{2}\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \left({k}−\mathrm{1}\right)!\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\frac{{k}}{\mathrm{2}}} {cos}\left({k}\:{arctan}\left(\frac{\mathrm{1}}{{x}}\right)\right)}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)^{{k}} }\:\Rightarrow \\ $$$${f}^{\left({n}\right)} \left({x}\right)=\left(−\mathrm{2}\right)^{{n}} {e}^{−\mathrm{2}{x}} {ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right) \\ $$$$+\sum_{{k}=\mathrm{1}} ^{{n}} \:{C}_{{n}} ^{{k}} \:\:\frac{\mathrm{2}\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \left({k}−\mathrm{1}\right)!\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\frac{{k}}{\mathrm{2}}} {cos}\left({karctan}\left(\frac{\mathrm{1}}{{x}}\right)\right)}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)^{{k}} }\left(−\mathrm{2}\right)^{{n}−{k}} {e}^{−\mathrm{2}{x}} \\ $$
Commented by mathmax by abdo last updated on 09/Sep/19
f^((n)) (0) =2Σ_(k=1) ^(n ) C_n ^k (−1)^(k−1) (k−1)!cos(((kπ)/2))(−2)^(n−k) f(x) =f(0) +Σ_(n=1) ^∞ ((f^((n)) (0))/(n!)) x^n =Σ_(n=1) ^∞ (2/(n!))(Σ_(k=1) ^n C_n ^k (−1)^(k−1) (k−1)!(−2)^(n−k) cos(((kπ)/2)))x^n
$${f}^{\left({n}\right)} \left(\mathrm{0}\right)\:=\mathrm{2}\sum_{{k}=\mathrm{1}} ^{{n}\:} \:{C}_{{n}} ^{{k}} \:\:\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \left({k}−\mathrm{1}\right)!{cos}\left(\frac{{k}\pi}{\mathrm{2}}\right)\left(−\mathrm{2}\right)^{{n}−{k}} \\ $$$${f}\left({x}\right)\:={f}\left(\mathrm{0}\right)\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{f}^{\left({n}\right)} \left(\mathrm{0}\right)}{{n}!}\:{x}^{{n}} \\ $$$$=\sum_{{n}=\mathrm{1}} ^{\infty} \frac{\mathrm{2}}{{n}!}\left(\sum_{{k}=\mathrm{1}} ^{{n}} \:{C}_{{n}} ^{{k}} \left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \left({k}−\mathrm{1}\right)!\left(−\mathrm{2}\right)^{{n}−{k}} \:{cos}\left(\frac{{k}\pi}{\mathrm{2}}\right)\right){x}^{{n}} \\ $$

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