Why we should call this : $a,b,c,d,\cdots$ sequence better than calling it series with $a,b,c,d,\cdots$ are integers?

Question

I'm confused why mathematical community called $a,b,c,d,\cdots$ a sequence better than calling it a series with $a,b,c,d,\cdots$ are integers , In addition to that they provided interactive website for Sloan $2008$ under this name " Encyclopedia of integer sequence however terms of sequence here are infinite by the way it satisfies mathematical notion of series better than calling it sequence , Then Why community considered that ?

Answer 1

You may have a slight confusion about what the terms actually mean.

A sequence $(a_n)$ is a ordered countable collection of elements. In the context of $a_n \in \mathbb{R}$, for example, we can define a sequence of partial sums, $$ s_n = \sum_{k=n_0}^n a_n $$ and this new sequence is called the series of the sequence $a_n$.

So given some numbers $(b_n)$, it is natural to associate them with a sequence. You can also associate them with a series, in which case you need to find the sequence $a_n$ for which $b_n$ will be partial sums...

Answer 2

In everyday language "sequence" and "series" are often interchangeable. When mathematicians use words from everyday language to do mathematics they provide them with precise mathematical definitions. We happen to have agreed on "sequence" for $$ a, b, c, \ldots $$ and "series" for $$ a + b + c + \cdots . $$ So Sloane's encyclopedia is an encyclopedia of sequences.

Your question suggests that you'd have preferred the other choice, but that's not going to happen.

By analogy, in everyday language "or" sometimes is "or but not both" and sometimes it's "one or both". Mathematicians have decided that in technical work it always means the latter.