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Question Number 126179 by mathmax by abdo last updated on 17/Dec/20
let f(x)= e^(−2x) actan (3x+1) 1)calculste f^((n)) (x) and f^((n)) (0) 2) if f(x)=Σ a_n x^n determine the sequence a_n 3) calculate ∫_0 ^∞ f(x)dx
$$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)=\:\mathrm{e}^{−\mathrm{2x}} \:\mathrm{actan}\:\left(\mathrm{3x}+\mathrm{1}\right) \\ $$$$\left.\mathrm{1}\right)\mathrm{calculste}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{x}\right)\:\mathrm{and}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:\mathrm{if}\:\mathrm{f}\left(\mathrm{x}\right)=\Sigma\:\mathrm{a}_{\mathrm{n}} \mathrm{x}^{\mathrm{n}} \:\mathrm{determine}\:\mathrm{the}\:\mathrm{sequence}\:\mathrm{a}_{\mathrm{n}} \\ $$$$\left.\mathrm{3}\right)\:\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} \\ $$
Answered by mathmax by abdo last updated on 19/Dec/20
1) by lebniz formulae f^((n)) (x)=Σ_(k=0) ^n C_n ^k (arctan(3x+1))^((k)) (e^(−2x) )^()n−k)) =C_n ^o arctan(3x+1)(−2)^n e^(−2x) +Σ_(k=1) ^n C_n ^k (arctan(3x+1))^((k)) (−2)^(n−k) e^(−2x) let calculate (arctan(3x+1))^k we have (dx/dx)(arctan(3x+1))=(3/(1+(3x+1)^2 )) ⇒(arctan(3x+1))^((k)) =3((1/((3x+1)^2 +1)))^((k−1)) =3((1/((3x+1+i)(3x+1−i))))^((k−1)) =(1/3)((1/(x+((1+i)/3))(x+((1−i)/3)))))^((k−1)) =(1/(3(((1−i)/3)−((1+i)/3))))((1/(x+((1+i)/3)))−(1/(x+((1−i)/3))))^((k−1)) =−(1/(2i))((1/(x+((√2)/3)e^((iπ)/4) ))−(1/(x+((√2)/3)e^(−((iπ)/4)) )))^((k−1)) =(i/2){ (((−1)^(k−1) (k−1)!)/((x+((√2)/3)e^((iπ)/4) )^k ))−(((−1)^(k−1) (k−1)!)/((x+((√2)/3)e^(−((iπ)/4)) )^k ))} =(i/2)(−1)^(k−1) (k−1)!{ ((−2i Im(x+((√2)/3)e^((iπ)/4) )^k )/((x+((√2)/3)e^((iπ)/4) )^k (x+((√2)/3)e^(−((iπ)/4)) )^k ))} ⇒ f^((n)) (x)=(−2)^n e^(−2x) arctan(3x+1)+Σ_(k=1) ^n C_n ^k (−2)^(n−k) e^(−2x) (−1)^k (k−1)!×((Im(x+((√2)/3)e^((iπ)/4) )^k )/((x+((√2)/3)e^((iπ)/4) )^k (x+((√2)/3)e^(−((iπ)/4)) )^k ))
$$\left.\mathrm{1}\right)\:\mathrm{by}\:\mathrm{lebniz}\:\mathrm{formulae}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{x}\right)=\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \:\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} \:\left(\mathrm{arctan}\left(\mathrm{3x}+\mathrm{1}\right)\right)^{\left(\mathrm{k}\right)} \left(\mathrm{e}^{−\mathrm{2x}} \right)^{\left.\right)\left.\mathrm{n}−\mathrm{k}\right)} \\ $$$$=\mathrm{C}_{\mathrm{n}} ^{\mathrm{o}} \:\:\mathrm{arctan}\left(\mathrm{3x}+\mathrm{1}\right)\left(−\mathrm{2}\right)^{\mathrm{n}} \:\mathrm{e}^{−\mathrm{2x}} \:+\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} \left(\mathrm{arctan}\left(\mathrm{3x}+\mathrm{1}\right)\right)^{\left(\mathrm{k}\right)} \left(−\mathrm{2}\right)^{\mathrm{n}−\mathrm{k}} \:\mathrm{e}^{−\mathrm{2x}} \\ $$$$\mathrm{let}\:\mathrm{calculate}\:\:\left(\mathrm{arctan}\left(\mathrm{3x}+\mathrm{1}\right)\right)^{\mathrm{k}} \:\:\mathrm{we}\:\mathrm{have} \\ $$$$\frac{\mathrm{dx}}{\mathrm{dx}}\left(\mathrm{arctan}\left(\mathrm{3x}+\mathrm{1}\right)\right)=\frac{\mathrm{3}}{\mathrm{1}+\left(\mathrm{3x}+\mathrm{1}\right)^{\mathrm{2}} }\:\Rightarrow\left(\mathrm{arctan}\left(\mathrm{3x}+\mathrm{1}\right)\right)^{\left(\mathrm{k}\right)} \:=\mathrm{3}\left(\frac{\mathrm{1}}{\left(\mathrm{3x}+\mathrm{1}\right)^{\mathrm{2}} +\mathrm{1}}\right)^{\left(\mathrm{k}−\mathrm{1}\right)} \\ $$$$=\mathrm{3}\left(\frac{\mathrm{1}}{\left(\mathrm{3x}+\mathrm{1}+\mathrm{i}\right)\left(\mathrm{3x}+\mathrm{1}−\mathrm{i}\right)}\right)^{\left(\mathrm{k}−\mathrm{1}\right)} \:=\frac{\mathrm{1}}{\mathrm{3}}\left(\frac{\mathrm{1}}{\left.\mathrm{x}+\frac{\mathrm{1}+\mathrm{i}}{\mathrm{3}}\right)\left(\mathrm{x}+\frac{\mathrm{1}−\mathrm{i}}{\mathrm{3}}\right)}\right)^{\left(\mathrm{k}−\mathrm{1}\right)} \\ $$$$=\frac{\mathrm{1}}{\mathrm{3}\left(\frac{\mathrm{1}−\mathrm{i}}{\mathrm{3}}−\frac{\mathrm{1}+\mathrm{i}}{\mathrm{3}}\right)}\left(\frac{\mathrm{1}}{\mathrm{x}+\frac{\mathrm{1}+\mathrm{i}}{\mathrm{3}}}−\frac{\mathrm{1}}{\mathrm{x}+\frac{\mathrm{1}−\mathrm{i}}{\mathrm{3}}}\right)^{\left(\mathrm{k}−\mathrm{1}\right)} \\ $$$$=−\frac{\mathrm{1}}{\mathrm{2i}}\left(\frac{\mathrm{1}}{\mathrm{x}+\frac{\sqrt{\mathrm{2}}}{\mathrm{3}}\mathrm{e}^{\frac{\mathrm{i}\pi}{\mathrm{4}}} }−\frac{\mathrm{1}}{\mathrm{x}+\frac{\sqrt{\mathrm{2}}}{\mathrm{3}}\mathrm{e}^{−\frac{\mathrm{i}\pi}{\mathrm{4}}} }\right)^{\left(\mathrm{k}−\mathrm{1}\right)} \\ $$$$=\frac{\mathrm{i}}{\mathrm{2}}\left\{\:\:\frac{\left(−\mathrm{1}\right)^{\mathrm{k}−\mathrm{1}} \left(\mathrm{k}−\mathrm{1}\right)!}{\left(\mathrm{x}+\frac{\sqrt{\mathrm{2}}}{\mathrm{3}}\mathrm{e}^{\frac{\mathrm{i}\pi}{\mathrm{4}}} \right)^{\mathrm{k}} }−\frac{\left(−\mathrm{1}\right)^{\mathrm{k}−\mathrm{1}} \left(\mathrm{k}−\mathrm{1}\right)!}{\left(\mathrm{x}+\frac{\sqrt{\mathrm{2}}}{\mathrm{3}}\mathrm{e}^{−\frac{\mathrm{i}\pi}{\mathrm{4}}} \right)^{\mathrm{k}} }\right\} \\ $$$$=\frac{\mathrm{i}}{\mathrm{2}}\left(−\mathrm{1}\right)^{\mathrm{k}−\mathrm{1}} \left(\boldsymbol{\mathrm{k}}−\mathrm{1}\right)!\left\{\:\:\frac{−\mathrm{2i}\:\mathrm{Im}\left(\mathrm{x}+\frac{\sqrt{\mathrm{2}}}{\mathrm{3}}\mathrm{e}^{\frac{\mathrm{i}\pi}{\mathrm{4}}} \right)^{\mathrm{k}} }{\left(\mathrm{x}+\frac{\sqrt{\mathrm{2}}}{\mathrm{3}}\mathrm{e}^{\frac{\mathrm{i}\pi}{\mathrm{4}}} \right)^{\mathrm{k}} \left(\mathrm{x}+\frac{\sqrt{\mathrm{2}}}{\mathrm{3}}\mathrm{e}^{−\frac{\mathrm{i}\pi}{\mathrm{4}}} \right)^{\mathrm{k}} }\right\}\:\Rightarrow \\ $$$$\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{x}\right)=\left(−\mathrm{2}\right)^{\mathrm{n}} \:\mathrm{e}^{−\mathrm{2x}} \:\mathrm{arctan}\left(\mathrm{3x}+\mathrm{1}\right)+\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} \:\left(−\mathrm{2}\right)^{\mathrm{n}−\mathrm{k}} \:\mathrm{e}^{−\mathrm{2x}} \left(−\mathrm{1}\right)^{\mathrm{k}} \left(\mathrm{k}−\mathrm{1}\right)!×\frac{\mathrm{Im}\left(\mathrm{x}+\frac{\sqrt{\mathrm{2}}}{\mathrm{3}}\mathrm{e}^{\frac{\mathrm{i}\pi}{\mathrm{4}}} \right)^{\mathrm{k}} }{\left(\mathrm{x}+\frac{\sqrt{\mathrm{2}}}{\mathrm{3}}\mathrm{e}^{\frac{\mathrm{i}\pi}{\mathrm{4}}} \right)^{\mathrm{k}} \left(\mathrm{x}+\frac{\sqrt{\mathrm{2}}}{\mathrm{3}}\mathrm{e}^{−\frac{\mathrm{i}\pi}{\mathrm{4}}} \right)^{\mathrm{k}} } \\ $$$$ \\ $$
Commented by mathmax by abdo last updated on 19/Dec/20
f^((n)) (0)=(−2)^n (π/4) +Σ_(k=1) ^n (−2)^(n−k) C_n ^k (−1)^k (k−1)!×(((((√2)/3))^k sin(((kπ)/4)))/((((√2)/3))^k )) f^((n)) (0)=(π/4)(−2)^n +Σ_(k=1) ^n (−1)^k (−2)^(n−k) C_n ^k (k−1)!sin(((kπ)/4))
$$\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{0}\right)=\left(−\mathrm{2}\right)^{\mathrm{n}} \frac{\pi}{\mathrm{4}}\:+\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\left(−\mathrm{2}\right)^{\boldsymbol{\mathrm{n}}−\boldsymbol{\mathrm{k}}} \:\boldsymbol{\mathrm{C}}_{\mathrm{n}} ^{\mathrm{k}} \:\left(−\mathrm{1}\right)^{\mathrm{k}} \left(\mathrm{k}−\mathrm{1}\right)!×\frac{\left(\frac{\sqrt{\mathrm{2}}}{\mathrm{3}}\right)^{\mathrm{k}} \mathrm{sin}\left(\frac{\mathrm{k}\pi}{\mathrm{4}}\right)}{\left(\frac{\sqrt{\mathrm{2}}}{\mathrm{3}}\right)^{\mathrm{k}} } \\ $$$$\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{0}\right)=\frac{\pi}{\mathrm{4}}\left(−\mathrm{2}\right)^{\mathrm{n}} \:+\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\left(−\mathrm{1}\right)^{\mathrm{k}} \left(−\mathrm{2}\right)^{\mathrm{n}−\mathrm{k}} \:\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} \left(\mathrm{k}−\mathrm{1}\right)!\mathrm{sin}\left(\frac{\mathrm{k}\pi}{\mathrm{4}}\right) \\ $$$$ \\ $$
Commented by mathmax by abdo last updated on 19/Dec/20
2) f(x)=Σ a_n x^n ⇒a_n =((f^((n)) (0))/(n!)) and f^((n)) (0) is known
$$\left.\mathrm{2}\right)\:\:\mathrm{f}\left(\mathrm{x}\right)=\Sigma\:\mathrm{a}_{\mathrm{n}} \mathrm{x}^{\mathrm{n}} \:\Rightarrow\mathrm{a}_{\mathrm{n}} =\frac{\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{0}\right)}{\mathrm{n}!}\:\:\mathrm{and}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{0}\right)\:\mathrm{is}\:\mathrm{known} \\ $$
Commented by mathmax by abdo last updated on 19/Dec/20
sorry f^((n)) (0) =(π/4)(−2)^n +Σ_(k=1) ^n (−2)^(n−k) C_n ^k (−1)^k (k−1)!((3/( (√2))))^k sin(((kπ)/4))
$$\mathrm{sorry}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{0}\right)\:=\frac{\pi}{\mathrm{4}}\left(−\mathrm{2}\right)^{\mathrm{n}} \:+\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\left(−\mathrm{2}\right)^{\mathrm{n}−\mathrm{k}} \:\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} \:\left(−\mathrm{1}\right)^{\mathrm{k}} \left(\mathrm{k}−\mathrm{1}\right)!\left(\frac{\mathrm{3}}{\:\sqrt{\mathrm{2}}}\right)^{\mathrm{k}} \:\mathrm{sin}\left(\frac{\mathrm{k}\pi}{\mathrm{4}}\right) \\ $$

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