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Question Number 35766 by abdo mathsup 649 cc last updated on 23/May/18
let f(x)=arctan(1+x) and f_n (x)= e^(−nx) 1) define fof_n (x) and f_n of(x) 2) study the nature of the series Σ_(n=0) ^(+∞) fof_n (x) and Σ_(n=0) ^∞ f_n of(x) .
$${let}\:{f}\left({x}\right)={arctan}\left(\mathrm{1}+{x}\right)\:\:{and}\:\:{f}_{{n}} \left({x}\right)=\:{e}^{−{nx}} \\ $$$$\left.\mathrm{1}\right)\:{define}\:{fof}_{{n}} \left({x}\right)\:{and}\:{f}_{{n}} {of}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{study}\:{the}\:{nature}\:{of}\:{the}\:{series}\:\sum_{{n}=\mathrm{0}} ^{+\infty} {fof}_{{n}} \left({x}\right) \\ $$$${and}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{f}_{{n}} {of}\left({x}\right)\:. \\ $$
Commented by maxmathsup by imad last updated on 02/Sep/18
1) we have fof_n (x)=f(f_n (x))=arctan(1+f_n (x))=arctan(1+e^(−nx) ) f_n 0f(x) = f_n (f(x))=e^(−nf(x)) =e^(−narctan(1+x)) 2)Σ_(n=0) ^∞ fof_n (x) =Σ_(n=0) ^∞ arctan(1+e^(−nx) ) =Σ u_n (x) if x<0 lim_(n→+∞) arctan(1+e^(−nx) ) ≠0 so Σ u_n (x)diverges if x>0 u_n (x)∼ e^(−nx) (n→+∞) but Σ e^(−nx) converges so the convergence simple of Σ u_n (x) is assured.also Σ f_n of = Σ_n e^(−n arctan(1+x)) =Σ_n (e^(−arctan(1+x)) )^n ⇒ if arctan(1+x)>0 the serie converges if arctan(1+x)<0 the serie diverges.
$$\left.\mathrm{1}\right)\:{we}\:{have}\:{fof}_{{n}} \left({x}\right)={f}\left({f}_{{n}} \left({x}\right)\right)={arctan}\left(\mathrm{1}+{f}_{{n}} \left({x}\right)\right)={arctan}\left(\mathrm{1}+{e}^{−{nx}} \right) \\ $$$${f}_{{n}} \mathrm{0}{f}\left({x}\right)\:=\:{f}_{{n}} \left({f}\left({x}\right)\right)={e}^{−{nf}\left({x}\right)} ={e}^{−{narctan}\left(\mathrm{1}+{x}\right)} \\ $$$$\left.\mathrm{2}\right)\sum_{{n}=\mathrm{0}} ^{\infty} \:{fof}_{{n}} \left({x}\right)\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:{arctan}\left(\mathrm{1}+{e}^{−{nx}} \right)\:=\Sigma\:{u}_{{n}} \left({x}\right) \\ $$$${if}\:{x}<\mathrm{0}\:\:{lim}_{{n}\rightarrow+\infty} {arctan}\left(\mathrm{1}+{e}^{−{nx}} \right)\:\neq\mathrm{0}\:\:{so}\:\Sigma\:{u}_{{n}} \left({x}\right){diverges} \\ $$$${if}\:{x}>\mathrm{0}\:\:{u}_{{n}} \left({x}\right)\sim\:{e}^{−{nx}} \:\:\:\left({n}\rightarrow+\infty\right)\:\:{but}\:\Sigma\:{e}^{−{nx}} \:{converges}\:{so}\:{the}\:{convergence} \\ $$$${simple}\:{of}\:\Sigma\:{u}_{{n}} \left({x}\right)\:{is}\:{assured}.{also} \\ $$$$\Sigma\:{f}_{{n}} {of}\:\:=\:\sum_{{n}} \:{e}^{−{n}\:{arctan}\left(\mathrm{1}+{x}\right)} \:\:\:=\sum_{{n}} \:\:\:\:\left({e}^{−{arctan}\left(\mathrm{1}+{x}\right)} \right)^{{n}} \:\Rightarrow \\ $$$${if}\:{arctan}\left(\mathrm{1}+{x}\right)>\mathrm{0}\:\:{the}\:{serie}\:{converges} \\ $$$${if}\:{arctan}\left(\mathrm{1}+{x}\right)<\mathrm{0}\:{the}\:{serie}\:{diverges}. \\ $$

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