Menu Close

1-2-1-5-2-3-2-2-9-2-5-2-3-13-2-7-2-4-17-2-

Tinku Tara 0 Comments
Pin




Question Number 123807 by Dwaipayan Shikari last updated on 28/Nov/20
((((1/2))!(1)!)/(((5/2))!))−((((3/2))!(2)!)/(((9/2))!))+((((5/2))!3!)/((((13)/2))!))−((((7/2))!4!)/((((17)/2))!))+....
$$\frac{\left(\frac{\mathrm{1}}{\mathrm{2}}\right)!\left(\mathrm{1}\right)!}{\left(\frac{\mathrm{5}}{\mathrm{2}}\right)!}−\frac{\left(\frac{\mathrm{3}}{\mathrm{2}}\right)!\left(\mathrm{2}\right)!}{\left(\frac{\mathrm{9}}{\mathrm{2}}\right)!}+\frac{\left(\frac{\mathrm{5}}{\mathrm{2}}\right)!\mathrm{3}!}{\left(\frac{\mathrm{13}}{\mathrm{2}}\right)!}−\frac{\left(\frac{\mathrm{7}}{\mathrm{2}}\right)!\mathrm{4}!}{\left(\frac{\mathrm{17}}{\mathrm{2}}\right)!}+…. \\ $$
Commented by Dwaipayan Shikari last updated on 28/Nov/20
Σ_(n=1) ^∞ (−1)^(n+1) ((Γ(n+(1/2))Γ(n+1))/(Γ(2n+(3/2)))) =Σ_(n=1) ^∞ (−1)^(n+1) β(n+(1/2),n+1) =∫_0 ^1 Σ_(n=1) ^∞ (−1)^(n+1) x^(n−(1/2)) (1−x)^n dx =∫_0 ^1 (1/( (√x)))Σ_(n=1) ^∞ (−1)^(n+1) x^n (1−x)^n dx =∫_0 ^1 (1/( (√x))).(1/(1+x(1−x)))dx x=t^2 ⇒1=2t(dt/dx) =2∫_0 ^1 (1/(1+t^2 −t^4 ))dx =−2∫_0 ^1 (1/(t^4 −t^2 −1))dt =−2∫_0 ^1 (1/(t^2 −((√((1+(√3))/2)))^2 ))+2∫_0 ^1 (1/(t^2 −((√((1−(√3))/2)))^2 )) =(2/( (√((1−(√3))/2))))log(((t−(√((1−(√3))/2)))/(t+(√((1−(√3))/2)))))−(2/( (√((1+(√3))/2))))log(((t−(√((1+(√3))/2)))/(t+(√((1+(√3))/2)))))
$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} \frac{\Gamma\left({n}+\frac{\mathrm{1}}{\mathrm{2}}\right)\Gamma\left({n}+\mathrm{1}\right)}{\Gamma\left(\mathrm{2}{n}+\frac{\mathrm{3}}{\mathrm{2}}\right)} \\ $$$$=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} \beta\left({n}+\frac{\mathrm{1}}{\mathrm{2}},{n}+\mathrm{1}\right) \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} {x}^{{n}−\frac{\mathrm{1}}{\mathrm{2}}} \left(\mathrm{1}−{x}\right)^{{n}} {dx} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\:\sqrt{{x}}}\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} {x}^{{n}} \left(\mathrm{1}−{x}\right)^{{n}} {dx} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\:\sqrt{{x}}}.\frac{\mathrm{1}}{\mathrm{1}+{x}\left(\mathrm{1}−{x}\right)}{dx}\:\:\:\:\:\:\:\:\:\:{x}={t}^{\mathrm{2}} \Rightarrow\mathrm{1}=\mathrm{2}{t}\frac{{dt}}{{dx}} \\ $$$$=\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\mathrm{1}+{t}^{\mathrm{2}} −{t}^{\mathrm{4}} }{dx}\:=−\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{{t}^{\mathrm{4}} −{t}^{\mathrm{2}} −\mathrm{1}}{dt} \\ $$$$=−\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{{t}^{\mathrm{2}} −\left(\sqrt{\frac{\mathrm{1}+\sqrt{\mathrm{3}}}{\mathrm{2}}}\right)^{\mathrm{2}} }+\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{{t}^{\mathrm{2}} −\left(\sqrt{\frac{\mathrm{1}−\sqrt{\mathrm{3}}}{\mathrm{2}}}\right)^{\mathrm{2}} } \\ $$$$=\frac{\mathrm{2}}{\:\sqrt{\frac{\mathrm{1}−\sqrt{\mathrm{3}}}{\mathrm{2}}}}{log}\left(\frac{{t}−\sqrt{\frac{\mathrm{1}−\sqrt{\mathrm{3}}}{\mathrm{2}}}}{{t}+\sqrt{\frac{\mathrm{1}−\sqrt{\mathrm{3}}}{\mathrm{2}}}}\right)−\frac{\mathrm{2}}{\:\sqrt{\frac{\mathrm{1}+\sqrt{\mathrm{3}}}{\mathrm{2}}}}{log}\left(\frac{{t}−\sqrt{\frac{\mathrm{1}+\sqrt{\mathrm{3}}}{\mathrm{2}}}}{{t}+\sqrt{\frac{\mathrm{1}+\sqrt{\mathrm{3}}}{\mathrm{2}}}}\right) \\ $$$$ \\ $$

Pin

Leave a Reply

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