I have the product $$\prod_{k=0}^n (1+a_k)$$ Does this product have a special name under which I can find some of its properties? I appreciate any reference for this product.
Note: Because of the inequality $1+x \le e^x$ I have already found the inequality $$\prod_{k=0}^n (1+a_k) \le \exp(\sum_{k=0}^n a_k)$$ if $\forall k \in \{0\ldots n\}: a_k \ge 0$.
Since given a product $\prod_i a_i$ setting $a_i' = a_i - 1$ will transform it in a product $\prod_i (1 + a_i')$ there is nothing special about that type of product without further restrictions.
That being said, the form you gave is common when considering infinite products, among others due to the inequality you mentioned that helps to establish that $\prod_{i=1}^{\infty} (1 + a_i)$ converges if and only if $\sum_{i=1}^{\infty} a_i$ converges.