Why is it useful to define sequences as being infinite? In every Real Analysis text that I've seen, the word 'sequence' always means "an infinite list of real numbers". Why is it so useful that the definition includes "infinite"? I have a similar issue with anything that doesn't distinguish between infinite series and finite series. 
 A: This is a soft question so I will answer it this way. This is largely based on my personal experience.
First, when people say sequence, they really mean a mapping from the set of all natural numbers to the real numbers. That is, for any $n\in\mathbb{R}$, there exists a corresponding real number $x_n$. Since there are infinitely many natural numbers, sequence has to be infinite in this sense.
But wait a second, it makes perfect sense talking about "finite sequence"! You just restrict your attention on a finite subset of $\mathbb{N}$, for example, $\{0,1,2,3,\cdots,27\}$. But you don't hear people talking about these kind of sequences often. There are some reasons behind it.
In real analysis, at least in undergraduate, infinite sequence really is the core topic. First, analysis is the study of limit, in some sense. When studying limit, sequence is a powerful tool that we can use to understand more complicated topics simply because sequence is the simplest object out there. Most textbooks start by talking about sequence, then you use sequence to study series and functions. 
If you think about it, series are complicated things: how the heck can we understand an infinite sum? After all, what is an infinite sum? Does it even make sense? By defining partial sum of a series, we can basically treat series as limit of sequences. Functions are complicated objects, too. They can be continuous or discontinuous, differentiable or nondifferentiable. By defining various properties of functions using sequence, we are basically reducing an unknown problem and unknown object to a known, well-studied object.
A: I am not a historian, but I think it's reasonable to assume the concept of sequences developed initially in connection with the concept of approximation. Take the area of a circle, for example. Take regular polygons with $n$ sides, inscribed in the circle. The area of each polygon is an approximation for the area of the circle, but to make real sense of this, you must allow $n$ to run through all natural numbers, you can't just limit it to finitely many $n$'s.
