I have become interested in taking the $n^{th}$ term of a series and evaluating a product whose $n^{th}$ term is $(1+a_n)$. After looking around I came across the following inequality:
$\sum_{n=0}^\infty a_n \leq \prod_{n=0}^\infty (1+a_n)\leq \exp(\sum_{n=0}^\infty a_n)$
I was wondering if anyone had a proof, or could offer any suggestions of a proof, for this inequality. By assuming $a_k>0$, I have managed to prove that the product is less then the exponential using $1+a_k \leq e^{a_k}$.
Thanks for the help.