I ran across this confounding limit I am wondering about. It is as follows:
$\displaystyle \lim_{n\to \infty}\frac{1-(1-c^{n})^{2n}}{(1-c)^{2n}}, \;\ 0
I played around with this on Maple and found that if c is less than approximately .382 (but
greater than 0), it converges to 0. If c is greater than .382 (but less than 1), it
diverges. What is it about .382?.
.382 is an approximation. By playing around more I could have taken it out to more decimal places.
The actual problem asks to prove that the above limit is < $\frac{1}{p(n)}$, where p(n) is a polynomial.
I was mainly wondering how to solve the limit and why .382 is so significant.
Thank you all very much. You are always a big help.