I am wanting to prove that
$$\left|\ln\left(1+\sum_{i=1}^{2^n-1} (a_i(n))^n\right)\right|= O(n \ln(2))$$ where $a_i(n)$ are functions of $n\in \Bbb{Z}$ bounded such that $|a_i(n)|\lt 1\; \forall n$. The prove is trivial if we know that $a_i(n) \gt 0$ since then $1\le 1+\sum_i (a_i(n))^n \le2^n$. But how would I go about proving it for where in general $a_i(n)$ can be negative (if it is indeed true)?