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Question Number 83245 by mathmax by abdo last updated on 29/Feb/20
let f(x) =e^(−2x) ln(1+2x) 1) find f^((n)) (x) and f^((n)) (0) 2)developp f at integr serie
$${let}\:{f}\left({x}\right)\:={e}^{−\mathrm{2}{x}} {ln}\left(\mathrm{1}+\mathrm{2}{x}\right) \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}^{\left({n}\right)} \left({x}\right)\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right){developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$
Commented by mathmax by abdo last updated on 03/Mar/20
1) f^((n)) (x)=Σ_(k=0) ^n C_n ^k (ln(2x+1))^((k)) (e^(−2x) )^((n−k)) =ln(2x+1)(−2)^n e^(−2x) +Σ_(k=1) ^n C_n ^k (ln(2x+1))^((k)) (−2)^(n−k) e^(−2x) we have (ln(2x+1))^((1)) =(2/(2x+1)) =(1/(x+(1/2))) ⇒ (ln(2x+1))^((k)) =((1/(x+(1/2))))^((k−1)) =(((−1)^(k−1) (k−1)!)/((x+(1/2))^k )) ⇒ f^((n)) (x)=ln(2x+1)(−2)^n e^(−2x) +Σ_(k=1) ^n C_n ^k ((2^k (−1)^(k−1) (k−1)!)/((2x+1)^k ))(−2)^(n−k) e^(−2x) =ln(2x+1)(−2)^n e^(−2x) +2^n Σ_(k=1) ^n C_n ^k (((−1)^(k−1+n−k) (k−1)!)/((2x+1)^k )) e^(−2x) f^((n)) (x)=(−2)^n e^(−2x) { ln(2x+1)+Σ_(k=1) ^n (k−1)! ×(C_n ^k /((2x+1)^k ))} f^((n)) (0) =Σ_(k=1) ^n (k−1)!C_n ^k =Σ_(k=1) ^n (k−1)!×((n!)/(k!(n−k)!)) =n!Σ_(k=1) ^n (1/(k(n−k)!))
$$\left.\mathrm{1}\right)\:{f}^{\left({n}\right)} \left({x}\right)=\sum_{{k}=\mathrm{0}} ^{{n}} {C}_{{n}} ^{{k}} \:\left({ln}\left(\mathrm{2}{x}+\mathrm{1}\right)\right)^{\left({k}\right)} \left({e}^{−\mathrm{2}{x}} \right)^{\left({n}−{k}\right)} \\ $$$$={ln}\left(\mathrm{2}{x}+\mathrm{1}\right)\left(−\mathrm{2}\right)^{{n}} \:{e}^{−\mathrm{2}{x}} \:+\sum_{{k}=\mathrm{1}} ^{{n}} \:{C}_{{n}} ^{{k}} \left({ln}\left(\mathrm{2}{x}+\mathrm{1}\right)\right)^{\left({k}\right)} \:\left(−\mathrm{2}\right)^{{n}−{k}} \:{e}^{−\mathrm{2}{x}} \\ $$$${we}\:{have}\:\left({ln}\left(\mathrm{2}{x}+\mathrm{1}\right)\right)^{\left(\mathrm{1}\right)} =\frac{\mathrm{2}}{\mathrm{2}{x}+\mathrm{1}}\:=\frac{\mathrm{1}}{{x}+\frac{\mathrm{1}}{\mathrm{2}}}\:\Rightarrow \\ $$$$\left({ln}\left(\mathrm{2}{x}+\mathrm{1}\right)\right)^{\left({k}\right)} =\left(\frac{\mathrm{1}}{{x}+\frac{\mathrm{1}}{\mathrm{2}}}\right)^{\left({k}−\mathrm{1}\right)} =\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \left({k}−\mathrm{1}\right)!}{\left({x}+\frac{\mathrm{1}}{\mathrm{2}}\right)^{{k}} }\:\Rightarrow \\ $$$${f}^{\left({n}\right)} \left({x}\right)={ln}\left(\mathrm{2}{x}+\mathrm{1}\right)\left(−\mathrm{2}\right)^{{n}} \:{e}^{−\mathrm{2}{x}} \:+\sum_{{k}=\mathrm{1}} ^{{n}} \:{C}_{{n}} ^{{k}} \:\:\frac{\mathrm{2}^{{k}} \left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \left({k}−\mathrm{1}\right)!}{\left(\mathrm{2}{x}+\mathrm{1}\right)^{{k}} }\left(−\mathrm{2}\right)^{{n}−{k}} \:{e}^{−\mathrm{2}{x}} \\ $$$$={ln}\left(\mathrm{2}{x}+\mathrm{1}\right)\left(−\mathrm{2}\right)^{{n}} \:{e}^{−\mathrm{2}{x}} \:+\mathrm{2}^{{n}} \sum_{{k}=\mathrm{1}} ^{{n}} \:{C}_{{n}} ^{{k}} \:\:\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}+{n}−{k}} \left({k}−\mathrm{1}\right)!}{\left(\mathrm{2}{x}+\mathrm{1}\right)^{{k}} }\:{e}^{−\mathrm{2}{x}} \\ $$$${f}^{\left({n}\right)} \left({x}\right)=\left(−\mathrm{2}\right)^{{n}} \:{e}^{−\mathrm{2}{x}} \left\{\:{ln}\left(\mathrm{2}{x}+\mathrm{1}\right)+\sum_{{k}=\mathrm{1}} ^{{n}} \left({k}−\mathrm{1}\right)!\:×\frac{{C}_{{n}} ^{{k}} }{\left(\mathrm{2}{x}+\mathrm{1}\right)^{{k}} }\right\} \\ $$$${f}^{\left({n}\right)} \left(\mathrm{0}\right)\:=\sum_{{k}=\mathrm{1}} ^{{n}} \left({k}−\mathrm{1}\right)!{C}_{{n}} ^{{k}} \:=\sum_{{k}=\mathrm{1}} ^{{n}} \:\left({k}−\mathrm{1}\right)!×\frac{{n}!}{{k}!\left({n}−{k}\right)!} \\ $$$$={n}!\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{{k}\left({n}−{k}\right)!} \\ $$
Commented by mathmax by abdo last updated on 03/Mar/20
2) f(x)=Σ_(n=0) ^∞ ((f^((n)) (0))/(n!)) x^n =f(0) +Σ_(n=1) ^∞ ((f^((n)) (0))/(n!))x^n =Σ_(n=1) ^∞ (Σ_(k=1) ^n (1/(k(n−k)!)))x^n
$$\left.\mathrm{2}\right)\:{f}\left({x}\right)=\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{f}^{\left({n}\right)} \left(\mathrm{0}\right)}{{n}!}\:{x}^{{n}} \:={f}\left(\mathrm{0}\right)\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{{f}^{\left({n}\right)} \left(\mathrm{0}\right)}{{n}!}{x}^{{n}} \\ $$$$=\sum_{{n}=\mathrm{1}} ^{\infty} \left(\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}\left({n}−{k}\right)!}\right){x}^{{n}} \\ $$

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