In a general definition, a sequence starts at zero or at one? In calculus textbooks we read that a sequence is a function whose domain is the set of positive integers. While in french textbooks we read that a sequence is a function whose domain is the set of non-negative integers. My question is about excluding zero from the definition of a sequence, what is the convention/reason behind that ? Thank you for your help!
 A: It doesn't matter. You simply have to choose a possible definition and use it consistently. Sequences don't even have to start at $0$ or $1$, they can start with any number and even be indexed by all integers $\mathbb{Z}$. There is also the more general notion of an indexed family.
A: Both are conventions, less a matter of reason than of custom. Logicians and set theorists count from $0$, as do many programming languages by default (C, Python, ...); other mathematicians prefer to count from $1$, and some programming languages do too by default (e.g. Basic, once upon a time). 
Anyone can justify their preference for one convention over the other. I prefer to count from $0$: it makes for a smoother set-theoretic development of the integers, rationals and reals; and counting from $1$ seems an accommodation of lingering medieval suspicions that $0$ isn't really a natural number. 
As @Dominik says in his answer, ultimately the choice doesn't matter (the resulting theorems are the same); what matters is to be clear and consistent about which convention you adhere to.
