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Question Number 142150 by iloveisrael last updated on 27/May/21

Π_(n=1) ^∞ (1+(1/n^4 )) =?

$$\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\:\left(\mathrm{1}+\frac{\mathrm{1}}{{n}^{\mathrm{4}} }\right)\:=?\: \\ $$

Commented by liberty last updated on 27/May/21

Π_(n=1) ^∞ (1+(1/n^4 ))=−((sin (((−1))^(1/4) π)sin ((((−1)^3 ))^(1/4) π))/π^2 )

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\left(\mathrm{1}+\frac{\mathrm{1}}{{n}^{\mathrm{4}} }\right)=−\frac{\mathrm{sin}\:\left(\sqrt[{\mathrm{4}}]{−\mathrm{1}}\:\pi\right)\mathrm{sin}\:\left(\sqrt[{\mathrm{4}}]{\left(−\mathrm{1}\right)^{\mathrm{3}} }\:\pi\right)}{\pi^{\mathrm{2}} } \\ $$

Answered by Dwaipayan Shikari last updated on 27/May/21

Π_(n=1) ^∞ (1+(1/n^4 ))=Π_(n=1) ^∞ (1+(((√i)/n))^2 )(1−(((√i)/n))^2 ) =((sinh((√i)π))/(4π^2 )).((sin((√i)π))/i)=−(((e^(((1+i)/( (√2)))π) −e^(−((1+i)/( (√2)))π) )(e^((i/( (√2)))−(1/( (√2)))π) −e^(−(i/( (√2)))+(1/( (√2)))π) ))/(4π^2 )) =−((e^((√2)iπ) −e^(−(√2)π) −e^(√(2π)) +e^(−(√2)iπ) )/(4π^2 ))=((cosh((√2)π)−cos((√2)π))/(2π^2 ))

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\left(\mathrm{1}+\frac{\mathrm{1}}{{n}^{\mathrm{4}} }\right)=\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\left(\mathrm{1}+\left(\frac{\sqrt{{i}}}{{n}}\right)^{\mathrm{2}} \right)\left(\mathrm{1}−\left(\frac{\sqrt{{i}}}{{n}}\right)^{\mathrm{2}} \right) \\ $$$$=\frac{{sinh}\left(\sqrt{{i}}\pi\right)}{\mathrm{4}\pi^{\mathrm{2}} }.\frac{{sin}\left(\sqrt{{i}}\pi\right)}{{i}}=−\frac{\left({e}^{\frac{\mathrm{1}+{i}}{\:\sqrt{\mathrm{2}}}\pi} −{e}^{−\frac{\mathrm{1}+{i}}{\:\sqrt{\mathrm{2}}}\pi} \right)\left({e}^{\frac{{i}}{\:\sqrt{\mathrm{2}}}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\pi} −{e}^{−\frac{{i}}{\:\sqrt{\mathrm{2}}}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\pi} \right)}{\mathrm{4}\pi^{\mathrm{2}} } \\ $$$$=−\frac{{e}^{\sqrt{\mathrm{2}}{i}\pi} −{e}^{−\sqrt{\mathrm{2}}\pi} −{e}^{\sqrt{\mathrm{2}\pi}} +{e}^{−\sqrt{\mathrm{2}}{i}\pi} }{\mathrm{4}\pi^{\mathrm{2}} }=\frac{{cosh}\left(\sqrt{\mathrm{2}}\pi\right)−{cos}\left(\sqrt{\mathrm{2}}\pi\right)}{\mathrm{2}\pi^{\mathrm{2}} } \\ $$

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