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Question Number 116626 by frc2crc last updated on 05/Oct/20

\underset{n = 1}{\prod^{\infty}} (\frac{a^{2} n^{2}}{{(a n)}^{2} - 1)}) = ? ? ?

Answered by mnjuly1970 last updated on 05/Oct/20

s o l u t i o n

\frac{s i n (π x)}{π x} = \underset{n =}{\prod^{\infty}} (1 - \frac{x^{2}}{n^{2}})

\frac{π x}{s i n (π x)} = \underset{n = 1}{\prod^{\infty}} \frac{n^{2}}{n^{2} - x^{2}}

x = \frac{1}{a} \Rightarrow \frac{π}{a s i n (\frac{π}{a})} = \underset{n = 1}{\prod^{\infty}} (\frac{a^{2} n^{2}}{a^{2} n^{2} - 1})

m . n . 1970

Answered by Dwaipayan Shikari last updated on 05/Oct/20

\underset{n = 1}{\prod^{\infty}} (1 - \frac{x^{2}}{n^{2}}) = \frac{s i n π x}{π x}

\prod^{\infty} (\frac{a^{2} n^{2}}{a^{2} n^{2} - 1}) = \frac{1}{\underset{n = 1}{\prod^{\infty}} (\frac{a^{2} n^{2} - 1}{a^{2} n^{2}})}

\frac{1}{\underset{n = 1}{\prod^{\infty}} (1 - \frac{1}{a^{2} n^{2}})} = \frac{1}{\frac{s i n \frac{π}{a}}{\frac{π}{a}}} = \frac{π}{a} c o s e c \frac{π}{a}