Menu Close

Question-189980




Question Number 189980 by sonukgindia last updated on 25/Mar/23
Answered by aleks041103 last updated on 26/Mar/23
I=∫_0 ^∞ x^(−((lnx)/n)) dx=∫_(−∞) ^( ∞) e^(−t(t/n)) e^t dt  x=e^t   I=∫_(−∞) ^( ∞) e^(−(t^2 /n −t)) dt  (t^2 /n)−t=((t/( (√n))))^2 −2(t/( (√n))) ((√n)/2) +(n/4)−(n/4)=  =((t/( (√n)))−((√n)/2))^2 −(n/4)  I=e^(n/4) ∫_(−∞) ^( ∞) e^(−((t/( (√n)))−((√n)/2))^2 ) dt=  =(√n)e^(n/4) ∫_(−∞) ^( ∞) e^(−t^2 ) dt=  =(√(nπ))e^(n/4)   ⇒S=Σ_(n=1) ^∞ ((√(nπ))/( (√(nπ))e^(n/4) ))=(1/e^(1/4) )Σ_(n=0) ^∞ (e^(−1/4) )^n =  =(1/e^(1/4) ) (1/(1−e^(−1/4) ))= (1/(e^(1/4) −1)) = S
$${I}=\int_{\mathrm{0}} ^{\infty} {x}^{−\frac{{lnx}}{{n}}} {dx}=\int_{−\infty} ^{\:\infty} {e}^{−{t}\frac{{t}}{{n}}} {e}^{{t}} {dt} \\ $$$${x}={e}^{{t}} \\ $$$${I}=\int_{−\infty} ^{\:\infty} {e}^{−\left({t}^{\mathrm{2}} /{n}\:−{t}\right)} {dt} \\ $$$$\frac{{t}^{\mathrm{2}} }{{n}}−{t}=\left(\frac{{t}}{\:\sqrt{{n}}}\right)^{\mathrm{2}} −\mathrm{2}\frac{{t}}{\:\sqrt{{n}}}\:\frac{\sqrt{{n}}}{\mathrm{2}}\:+\frac{{n}}{\mathrm{4}}−\frac{{n}}{\mathrm{4}}= \\ $$$$=\left(\frac{{t}}{\:\sqrt{{n}}}−\frac{\sqrt{{n}}}{\mathrm{2}}\right)^{\mathrm{2}} −\frac{{n}}{\mathrm{4}} \\ $$$${I}={e}^{{n}/\mathrm{4}} \int_{−\infty} ^{\:\infty} {e}^{−\left(\frac{{t}}{\:\sqrt{{n}}}−\frac{\sqrt{{n}}}{\mathrm{2}}\right)^{\mathrm{2}} } {dt}= \\ $$$$=\sqrt{{n}}{e}^{{n}/\mathrm{4}} \int_{−\infty} ^{\:\infty} {e}^{−{t}^{\mathrm{2}} } {dt}= \\ $$$$=\sqrt{{n}\pi}{e}^{{n}/\mathrm{4}} \\ $$$$\Rightarrow{S}=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\sqrt{{n}\pi}}{\:\sqrt{{n}\pi}{e}^{{n}/\mathrm{4}} }=\frac{\mathrm{1}}{{e}^{\mathrm{1}/\mathrm{4}} }\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left({e}^{−\mathrm{1}/\mathrm{4}} \right)^{{n}} = \\ $$$$=\frac{\mathrm{1}}{{e}^{\mathrm{1}/\mathrm{4}} }\:\frac{\mathrm{1}}{\mathrm{1}−{e}^{−\mathrm{1}/\mathrm{4}} }=\:\frac{\mathrm{1}}{{e}^{\mathrm{1}/\mathrm{4}} −\mathrm{1}}\:=\:{S} \\ $$
Commented by sonukgindia last updated on 26/Mar/23
thNks
$${thNks} \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *