When are two series the same?

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.

Solution 1:

Depends on your understanding of the term "series". If you are talking about the sequence of partial sums, then yes, two series are equal iff their respective terms are equal. If you are talking about the value of the series (suppose it converges), i.e. the limit of partial sums, then no, we can have two series with different terms that converge to the same value.

Solution 2:

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.

Solution 3:

As several others have stated, the question comes down to one's definition of "the same series". And that, in turn comes down to "what is a series?"

Speaking of "the set of sequences" is perfectly usual and unambiguous; a sequence is a special type of mapping, and one can speak of all such mappings. By contrast, "the set of series" sounds a little anomalous, but certainly suggests a series is its sequence of partial sums, and consequently two series are the same if and only if they have identical sequences of summands. (After all "the space of series" should be infinite dimensional; if a convergent series were identified with its sum, then the space of series would be one-dimensional.)

For posterity, here are some details: Let $(a_{k})_{k=0}^{\infty}$ be a sequence of real numbers, and let $(s_{n})_{n=0}^{\infty}$ be the sequence of partial sums, defined by $$ s_{n} = \sum_{k=0}^{n} a_{k},\quad n \geq 0, $$ namely, by the recursive specification $$ s_{0} = a_{0},\qquad s_{n+1} = s_{n} + a_{n+1}\quad\text{for $n \geq 0$.} \tag{*} $$

The sequence $(a_{k})_{k=0}^{\infty}$ is summable if the sequence of partial sums has a limit, in which case $s := \lim\limits_{n \to \infty} s_{n}$ is called the sum of the sequence $(a_{k})_{k=0}^{\infty}$ (or "the sum of the series $\sum_{k} a_{k}$").

A reasonable definition of "the same series" is equality of the sequences of partial sums,(1) namely $$ \sum_{k=0}^{\infty} a_{k}\quad\text{is the same series as}\quad \sum_{k=0}^{\infty} b_{k} $$ if and only if $$ \sum_{k=0}^{n} a_{k} = \sum_{k=0}^{n} b_{k}\quad\text{for all $n \geq 0$.} $$

Now, the partial sums of a sequence uniquely determine the terms of the sequence: Rearranging (*) gives $a_{0} = s_{0}$, and $a_{n+1} = s_{n+1} - s_{n}$ for $n \geq 0$. It follows that two series are "the same" in the preceding sense if and only if they have identical sequences of summands, if and only if $a_{k} = b_{k}$ for all $k \geq 0$.

If we accept this definition of "the same", then $1 + 2 + 3 + \dots$ and $0 + 1 + 2 + 3 + \dots$ are not the same series, because they do not have the same sequence of partial sums (or, equivalently, because they do not have identical summands).

1. A less reasonable definition of "the same series" — i.e., a definition less psychologically congruent with the underlying concept and therefore more prone to misunderstanding — is that the sums are equal. With this definition, $$ \sum_{k=0}^{\infty} \frac{(-1)^{k}}{(k+1) \log 2} \quad\text{and}\quad \sum_{k=0}^{\infty} \frac{1}{2^{k+1}} \quad\text{and}\quad \sum_{k=0}^{\infty} \frac{1}{e\, k!} \quad\text{and}\quad 1 + 0 + \dots + 0 + \cdots $$ (etc., etc.) are "the same series". (Doesn't that feel wrong, even putting aside that they converge at vastly different rates...?) Instead, this condition is better stated as "the series have the same sum/value" or "the sequences (of terms) have equal sums".

Solution 4:

Regarding your edited question:

Yes, it makes sense to talk about the set of all series, if the set is well-defined. Your question implies that your interest is in whether the set is well-ordered. Asking whether the set is well-ordered on the equality relation depends on a definition of equality.

For your purposes, perhaps one of the two definitions below makes sense makes sense:

Equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities (1) have the same value or (2) that the expressions represent the same mathematical object.

Based on (1), the set of all series S under the equality relation ({(S):=}), equality depends on the value of the sum of the series.

Based on (2), the two series you mention may not be considered equal, as they are not, strictly speaking, identical.

Solution 5:

Are the series $$ 0 + 1 + 2 + 3 + \dots \quad\text{and}\quad 1 + 2 + 3 + 4 + \dots $$ the same? What if a summation method (like "zeta-regularization") assigns different "sums" to them? Then certainly we would require that the two series be considered "different".

further explanation
First write $$ \frac{1}{1^s}+\frac{2}{2^s}+\frac{3}{3^s}+\frac{4}{4^s}+\dots =\sum_{n=1}^\infty \frac{n}{n^s} = \zeta(s-1) $$ which is true for $s>2$. But then plug in $s=0$, and say magic words "zeta regularization" to conclude $$ 1+2+3+4+\dots = \zeta(-1) = \frac{-1}{12} $$ Find this even in respected physics books!

Next, write $$ \frac{0}{1^s}+\frac{1}{2^s}+\frac{2}{3^s}+\frac{3}{4^s}+\dots= \sum_{n=1}^\infty \frac{n-1}{n^s} = \zeta(s-1)-\zeta(s) $$ which is true for $s>2$. But plug in $s=0$ to get $$ 0+1+2+3+4+\dots = \zeta(-1)-\zeta(0) = \frac{5}{12} $$

So surely the two series are different!