Let the binary operation $x*y=(x+y)/(1+xy)$ where $x,y\in (-1,1)$ Find $(1/2)*(1/3)*...*(1/1000)$
I found that $1/x*1/y=(x+y)/(1+xy)$ but I can not find a rule for $n$ such numbers and apply induction to calculate that sum.
Could you please tell me what would be another way to solve the problem?
HINT: To solve this problem, compute the product $(x_0)(x_1)...(x_n)$ of finitely many terms using your binary operation. Find the pattern, and prove what you see; It is a pretty symmetric result. After this, computing the final value should not be too hard. If you want a complete answer, then ask for one in the comments after you have tried it yourself.