Question and Answers Forum

All Questions      Topic List

Arithmetic Questions

Previous in All Question      Next in All Question      

Previous in Arithmetic      Next in Arithmetic      

Question Number 90561 by Tony Lin last updated on 24/Apr/20

prove that  Π_(n=2) ^∞ (1−(1/n^2 ))=(1/2)

$${prove}\:{that} \\ $$$$\underset{{n}=\mathrm{2}} {\overset{\infty} {\prod}}\left(\mathrm{1}−\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\right)=\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Commented by mathmax by abdo last updated on 24/Apr/20

let S_n =Π_(k=2) ^n (1−(1/k^2 )) ⇒ S_n =Π_(k=2) ^n ((k^2 −1)/k^2 )  =Π_(k=2) ^n  ((k−1)/k)×((k+1)/k) =Π_(k=2) ^n  ((k−1)/k)×Π_(k=2) ^n  ((k+1)/k)  =(1/2)×(2/3)×(3/4)×....((n−1)/n) ×(3/2)×(4/3)×....×(n/(n−1))×((n+1)/n)  =((n+1)/(2n)) ⇒lim_(n→+∞)  S_n =(1/2)

$${let}\:{S}_{{n}} =\prod_{{k}=\mathrm{2}} ^{{n}} \left(\mathrm{1}−\frac{\mathrm{1}}{{k}^{\mathrm{2}} }\right)\:\Rightarrow\:{S}_{{n}} =\prod_{{k}=\mathrm{2}} ^{{n}} \frac{{k}^{\mathrm{2}} −\mathrm{1}}{{k}^{\mathrm{2}} } \\ $$$$=\prod_{{k}=\mathrm{2}} ^{{n}} \:\frac{{k}−\mathrm{1}}{{k}}×\frac{{k}+\mathrm{1}}{{k}}\:=\prod_{{k}=\mathrm{2}} ^{{n}} \:\frac{{k}−\mathrm{1}}{{k}}×\prod_{{k}=\mathrm{2}} ^{{n}} \:\frac{{k}+\mathrm{1}}{{k}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}×\frac{\mathrm{2}}{\mathrm{3}}×\frac{\mathrm{3}}{\mathrm{4}}×....\frac{{n}−\mathrm{1}}{{n}}\:×\frac{\mathrm{3}}{\mathrm{2}}×\frac{\mathrm{4}}{\mathrm{3}}×....×\frac{{n}}{{n}−\mathrm{1}}×\frac{{n}+\mathrm{1}}{{n}} \\ $$$$=\frac{{n}+\mathrm{1}}{\mathrm{2}{n}}\:\Rightarrow{lim}_{{n}\rightarrow+\infty} \:{S}_{{n}} =\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Commented by Tony Lin last updated on 25/Apr/20

thanks sir

$${thanks}\:{sir} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com