TITLE: When are two series the same?
QUESTION [13 upvotes]: A series is an expression of the form
$$
\sum_{n=k}^{\infty} a_n
$$
where the $a_n$ are real numbers and they depend on $n$. If $a_n = b_n$ for all $n\geq k$, then I would assume that one would say that the two series
$$
\sum_{n=k}^{\infty} a_n\quad\text{and}\quad \sum_{n=k}^{\infty} b_n
$$
are the same series. Is it true that if two series
$$
\sum_{n=k}^{\infty} a_n\quad\text{and}\quad \sum_{n=k}^{\infty} b_n
$$
are the same, then $a_n = b_n$ are the same for all $n$?
The reason that I am asking is because I would think that the two series
$$
0 + 1 + 2 + 3 + \dots \quad\text{and}\quad 1 + 2 + 3 + 4 + \dots
$$
are the same, but $0\neq 1, 1\neq 2, \dots $.
So, when exactly are two series the same when considered as elements in the set of all series?
Edit: Given the comment and answer below, maybe what I need to know is if it makes sense to talk about the set of all series. If this does make sense, what does it mean that two elements of this set are the same.

REPLY [4 votes]: In order to answer OP's question we first have to clearly state the meaning(s) of the symbol
  \begin{align*}
  \sum_{n=k}^{\infty}a_n\tag{1}
  \end{align*}
  with $a_n$ being real numbers.

$$$$

First meaning: Infinite series are sequences of partial sums
When looking at a sequence $(a_k,a_{k+1},a_{k+2},\ldots) = (a_n)_{n\geq k}\,$
  we consider
  \begin{align*}
  (a_k,a_{k}+a_{k+1},a_{k}+a_{k+1}+a_{k+2},\ldots) = (\sum_{n=k}^{N}a_k)_{N\geq k}
  \end{align*}
  and define the infinite series (1) as the sequence of partial sums
  \begin{align*}
    \sum_{n=k}^{\infty}a_n:= (\sum_{n=k}^{N}a_k)_{N\geq k}\tag{2}
  \end{align*}

So, the LHS of (2) is only a new and very convenient representation for the sequence of partial sums
\begin{align*}
  (s_N)_{N\geq k}, \qquad s_N=\sum_{n=k}^{N}a_n\tag{3}
  \end{align*}

We conclude: Two (infinite) series
  \begin{align*}
  \sum_{n=k}^{\infty}a_n \quad\text{and}\quad \sum_{n=l}^{\infty}b_n
  \end{align*}
  are equal, iff the sequences of partial sums are equal, i.e. if $k=l$ and $a_n=b_n$ with $n\geq k$.

The symbol $\sum_{n=k}^{\infty}a_n$ has also a second meaning, namely

Second meaning: Infinite series are the limits of the sequences of partial sums
An infinite series $\sum_{n=k}^{\infty}a_n$ is said to be convergent, definitely divergent or indefinitely divergent according as the sequence of its partial sums shows the behaviour indicated by those names.
If, in the case of convergence, the sequence of partial sums, $s_N\rightarrow s$, then we say that $s$ is the value or the sum of the convergent infinite series and we write for brevity
\begin{align*}
  \sum_{n=k}^{\infty}a_n:=\lim_{N\rightarrow\infty}s_N=s
  \end{align*}
  so that $\sum_{n=k}^{\infty}a_n$ denotes not only the sequence $(s_N)_{N\geq k}$ but also the limit $s$, provided this limit exists.
We conclude:
  If two (infinite) series are used to represent the limit of a sequence of the partial sums they are equal iff both series converge and their limits are equal.


Notes:

Considering the argumentation above, we observe that there's no need to know anything about the set of all (infinite) series as a whole. We only need to precisely specify how an element is defined and then it's sufficient to compare any two of this kind to determine if they are equal.
Observe, that in (3) the real-valued sequence $(s_N)_{N\geq k}$ can also be written as real-valued function $s$ with domain $\mathbb{N}$ and codomain $\mathbb{R}$ mapping $N$ to $s_N$. So, in the first case we could also say, we are asking for equality of two functions.
In the second case, when we are asking for equality of convergent
infinite series, we are asking for equality of real numbers.


You can find this line of argumentation e.g. in K. Knopps Theory and application of infinite series.