Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 103436 by bramlex last updated on 15/Jul/20

$$\underset{\mathrm{n}=3}{\prod ^{\mathrm{\infty }}}(1-\frac{4}{{\mathrm{n}}^2})=?$$

Answered by Worm_Tail last updated on 15/Jul/20

$$\underset{3}{\prod ^{oo}}(1-\frac{4}{n^2})=a$$$$(1-\frac{4}{3^2})(1-\frac{4}{4^2})(1+\frac{4}{5^2})...=a$$$$ln((1-\frac{4}{3^2})(1-\frac{4}{4^2})(1+\frac{4}{5^2})...)=ln\left(a\right)$$$$ln(1-\frac{4}{3^2})+ln(1-\frac{4}{4^2})+ln(1+\frac{4}{5^2})...=ln\left(a\right)$$$$\underset{n=3}{\sum ^{oo}}\left(ln(1-\frac{4}{n^2})\right)=lna$$$$\underset{n=3}{\sum ^{oo}}\left(ln\left(\frac{(n+2)(n-2)}{n^2}\right)\right)=lna$$$$\underset{n=3}{\sum ^{oo}}(ln(n+2)+ln(n-2)-2ln\left(n\right))=lna$$$$\underset{n=3}{\sum ^{oo}}ln(n+2)+\underset{n=3}{\sum ^{oo}}ln(n-2)-2\underset{n=3}{\sum ^{oo}}ln\left(n\right)=lna$$$$\underset{n=3}{\sum ^{oo}}ln(n+2)+(ln\left(1\right)+ln\left(2\right)+ln\left(3\right)+ln\left(4\right)+\underset{n=7}{\sum ^{oo}}ln(n-2))-2(ln\left(3\right)+ln\left(4\right)+\underset{n=5}{\sum ^{oo}}ln\left(n\right))=lna$$$$\underset{n=3}{\sum ^{oo}}ln(n+2)=\underset{n=7}{\sum ^{oo}}ln(n-2)=\underset{n=5}{\sum ^{oo}}ln\left(n\right)=s$$$$s+(ln\left(1\right)+ln\left(2\right)+ln\left(3\right)+ln\left(4\right)+s)-2(ln\left(3\right)+ln\left(4\right)+s)=lna$$$$s+ln\left(1\right)+ln\left(2\right)+ln\left(3\right)+ln\left(4\right)+s-2ln\left(3\right)-2ln\left(4\right)-2s=lna$$$$s+ln\left(1\right)+ln\left(2\right)+ln\left(3\right)+ln\left(4\right)+s-2ln\left(3\right)-2ln\left(4\right)-2s=lna$$$$s+s-2s+ln\left(1\right)+ln\left(2\right)-ln\left(3\right)-ln\left(4\right)=ln\left(a\right)$$$$ln\left(\frac{2}{12}\right)=ln\left(a\right)$$$$\frac{2}{12}=a$$$$a=\frac{1}{6}$$

Commented by bobhans last updated on 15/Jul/20

$$waw..vialogarithm$$

Answered by bemath last updated on 15/Jul/20

Commented by bobhans last updated on 15/Jul/20

$$coollgraphic$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com