What's the difference between tuples and sequences? Both are ordered collections that can have repeated elements. Is there a difference? Are there other terms that are used for similar concepts, and how are these terms different?

Using a basic set theoretic definition, a tuple (a, b, c, ..) represents an element of the Cartesian product of sets  A x B x C ... 
In a vector space the tuple represents the components of a vector in terms of basis vectors.
A sequence on the other hand represents a function (usually of the natural numbers) to some set A, and strictly speaking a sequence is then a subset of N x A.
For numeric sequences, It makes sense to consider whether they are convergent. One could add the elements of a numeric sequence to get a series and consider if the corresponding series is convergent. I can't think of any equivalent concept for tuples even when they comprise numeric values.
