Formal Definition of a Series $\sum a_n$

Question

A series of real numbers can be defined like:

$$S: \mathbb{N} \to \mathbb{R}$$ $$\tag{1}\displaystyle{S_n = \sum_{i = 1}^{n} a_i}$$ $$\tag{2}S_n \to \displaystyle{\sum_{i = 1}^{\infty} a_i}$$

This is the most common definition of series. But my question is: Why start at $i = 1$? Or in other books, $i = 0$? In my opinion, doesn't matter what the initial term is. So, can this be an alternative definition:

$$\tag{3}\displaystyle{S_n = \sum_{i}^{n} a_i}; ~ i \in \mathbb{Z}, ~ i \leq n\quad ?$$

Answer 1

A series can start at any value - it's just conventional to start at $i=0$ or $i=1$, since you're summing a countably infinite number of terms, and the most obvious countably infinite set is $\mathbb{N}$ (the smallest element of which is either $0$ or $1$, depending on which convention you adopt).

Answer 2

Adoption of the convention of a series as a mapping $\mathbb{N} \to \mathbb{R}$ (or the codomain set of your choice) can be handled easily for cases where the series does not start at $i=1$. (We assume $0 \not \in \mathbb{N}$ for simplicity.)

Suppose we start at some $i > 1$, e.g. $\sum_{i=k}^\infty a_i$ for $k \ge 2$. Then we can "extend" the definition of $a_i$. Namely, define $$ \widetilde{a_i} := \begin{cases} a_i, & i \ge k \\ 0, & i < k \end{cases} $$ Then clearly $\sum_{i=k}^\infty a_i = \sum_{i=1}^\infty \widetilde{a_i}$.

Suppose the series starts at some $i < 1$, e.g. $\sum_{i=k}^\infty a_i$ for $k \le 0$. Then we can define $$ \widetilde{a_i} := a_{i-k+1} $$ and say $\sum_{i=k}^\infty a_i = \sum_{i=1}^\infty \widetilde{a_i}$.

What this shows us is that any series that has (finite) lower bound can be rephrased in terms of one that has lower bound $i=1$, so there's no fundamental issue in declaring the former to be series just the same.

Answer 3

Any indexing system can be used, as long as it's clear. For instance,

$S = \displaystyle{\sum_{x \in X} f(x)}$ can be valid, if $X$ is countable, and we have either absolute convergence or a clear canonical order.

We can also have something like $S = \displaystyle{\sum_{p : \text{ p is prime}} f(p)}$