What is a series?

Question

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?

Answer 1

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.

Answer 2

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 $$\sum_{n=1}^\infty a_n = \lim_{N\to\infty}\sum_{n=1}^Na_n$$

Answer 3

The question has nothing to do with infinity; you might just as well ask, what is a finite series, or, what is a polynomial?

The sum of an infinite series is the limit of its sequence of partial sums, but that doesn't mean that the infinite series is its sequence of partial sums. That's not the way the term "series" is used; it is the kind of nonsense you might get from a mathematics instructor who is being pressed for a formal definition.

The answer is that a series, like a polynomial, is a formal expression. We all know what a formal expression is; for most mathematical purposes there is no need to identify it with a particular set-theoretical object. What is a polynomial? You might choose to identify a polynomial (in one variable, over the real field) with its sequence of coefficients, and so define it as a function from the set of all nonnegative integers to the set of all real numbers which takes nonzero values at most a finite number of times. Or you could define a polynomial as a word in a certain language over an alphabet containing variables, numerals, $+$ and $-$ signs, etc.