Recent content by happyxiong531

Oh, I forget there is a condition that the sequence should satisfy that1-c\frac{a_1}{1+a_1}>0, so that all the elements in this sequence should be positive, and c cannot be too large. I have made some plots like c=0.5, c=2, the conclusion is correct. Thanks

Thank you for you reply. I think it's correct. First, I can have a_n*n\rightarrow 1 when c=1, from a_{n+1}=\frac{a_n}{1+a_n}=\frac{a_{n-1}}{1+2a_{n-1}}=\cdots=\frac{a_1}{1+(n+1)a_1} Then, let ca_n=b_n if c\neq 1, c is some constant. we can have...

I want to prove that if the sequence a_n satisfy that a_{n+1}=a_n\left(1-c\frac{a_n}{1+a_n}\right) then a_n*c*n\rightarrow 1 for all positive c. Like when c=1, then a_n*n\rightarrow 1, but if c\neq 1, it's difficult to prove.