isomorphism in a product

Question

$x*y=\frac{x+y}{1+xy} , x,y\in(-1,1).$ Calculate the value of $ \frac{1}{2}*\frac{1}{3}* \cdots *\frac{1}{1000}.$ I tried a lot of functions but I don't know how to find a good isomorphism and do the evaluation. Can somebody explain me how can I solve this and in which way to think,please? Not just giving me a function.

Answer 1

The function $\tanh:\Bbb R\to (-1;1) $ is a group isomorphism by addition formula $\tanh (\alpha+\beta)=\tanh (\alpha)\ast\tanh (\beta ) $. Your expression evaluates as \begin{align}\tanh\left (\sum_{n=2}^{1000}\tanh^{-1}\left (\frac 1n\right)\right) &=\tanh\left (\frac 12\log\left (\prod_{n=2}^{1000}\frac {1+1/n}{1-1/n}\right)\right)\\ &=\tanh\left (\frac 12\log\left (\prod_{n=2}^{1000}\frac {n+1}{n-1}\right)\right)\\ &=\tanh\left (\frac 12\log\left(\frac {1000\cdot 1001} {1\cdot 2}\right)\right)\\ &=\tanh\left (\frac 12\log (500500)\right)\\ &=\tanh\left (\tanh^{-1}\left(\frac {500501}{500499}\right)\right)\\ &=\frac {500501}{500499} \end{align}