Convergence of a Recursive Sequence: Proving a_n*c*n\rightarrow 1 for Positive c

[happyxiong531]

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.

 

[mathman]

What makes you believe it is true? Your question implies a_n*n ->\frac{1}{c}. Doesn't look right, especially for large c.

 

[happyxiong531]

What makes you believe it is true? Your question implies a_n*n -> 1/c. Doesn't look right, especially for large c.

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 b_{n+1}=b_n\left(1-\frac{b_n}{1+b_n/c}\right).

Actually, it's easy to prove a_n and b_n will go to zero,
so, \frac{b_n}{1+b_n/c}\sim\frac{b_n}{1+b_n}, thenb_n*n\rightarrow 1.

I have made a plot, it's correct no matter c is larger or less than 1.
But I think my proof is not strict.
Thanks for your concern.

 

[happyxiong531]

What makes you believe it is true? Your question implies a_n*n ->\frac{1}{c}. Doesn't look right, especially for large c.

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

 

[mathman]

Write out your complete proof.