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?

A series is the sequence of partial sums of another sequence.
The sum of a series is the limit of the sequence of partial sums, if it exists.
Just like you can write a sequence as $(a_n)$, it may help to write the corresponding series as $\sum a_n$. Note the absence of decoration in the sum sign.
The sum of a series is denoted $\displaystyle\sum_{n=1}^\infty a_n$ just as the limit of a sequence is denoted $\displaystyle\lim_{n\to\infty}a_n$.
The precise definition is $$\sum_{n=1}^\infty a_n = \lim_{N\to\infty}\sum_{n=1}^Na_n$$
