What is a series? This question is rather pedantic, but it is something that has been bothering me for some time.
Summing up infinitely many terms of a sequence is something that is done in pretty much every subfield of mathematics, so series are right at the core of mathematics. But strangely, I have never seen a formal definition of a series of the form "A series is...", whether I look in books on calculus or on Banach space theory. 
Also, the use of language seems somewhat inconsistent. Many texts formally define $\sum_{n=1}^\infty x_n$ to be $\lim_{N\to\infty}\sum_{n=1}^N x_n$ but then write something like "The series $\sum_{n=1}^\infty x_n$ converges if...", which would then mean "$\lim_{N\to\infty}\sum_{n=1}^N x_n$ converges if...", which makes no sense for then $\sum_{n=1}^\infty x_n$ is either a number (or a vector) or a meaningless expression such as "the largest natural number".
So what is the definition of a series? Or are series really just a way to speak about sequences and series do not exist as mathematical objects?

I think I remember that when I first learned about this, my professor said that this is the first 'abuse of notation' that we would encounter- the symbol $\sum_{n=0}^\infty a_n$ is both used for the sequence and its limit.
One way to answer your original question could be to think of a series as a pair of sequences $(a_n,b_n)$ such that $b_{n+1}-b_n=a_n$ and so make both the underlying sequence and the series to part of the data.
