Looking for a couple of sequence satisfying some conditions

Question

I am trying to solve the following problem:

Find $(c_n)_{n=1}^{\infty}$ and $(e_n)_{n=1}^{\infty}$ s.t.

0. $e_n>0$ for all n;
1. $c_n$ decreases to zero as n goes to infinity;
2. $c_{n+1}^{-e_n}>K$ for all n, where K is a constant strictly bigger than 1;
3. $\sum_{n=1}^{\infty} c_{n+1}^{-e_n}log(c_{n+1}^{-1})[c_n^{e_n}-c_{n+1}^{e_n}] < \infty$

I have seen that $c_n=\frac{1}{n}$ and $e_n=\frac{1}{n}$ satisfy point 1 and 3, while $c_{n+1}^{-e_n}$ goes to 1 as n goes to infinity. I have not been able to start from my example and obtain a different couple of sequences satisfying point 2 too.

Any idea?
Thank you in advance