# 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.

• What do you mean by the word "same"? I would call two series the "same" if their sequence of partial sums converge to the same value, but you might want to consider two series as the "same" only if $a_{n} \equiv b_{n}$ for all $n$. – JessicaK Apr 7 '15 at 16:16
• If $a_n=\left(\dfrac{-1}{2}\right)^n$ and $b_n = \left(\dfrac{1}{2}\right)^{2n+1}$ then $\displaystyle \sum_{n=0}^\infty a_n = \sum_{n=0}^\infty b_n$ but some people might suggest these are different series. – Henry Apr 7 '15 at 16:24
• Depends on what definition is most convenient to use, honestly. Is there a specific context? – Akiva Weinberger Apr 7 '15 at 16:34
• So, in summary: do no use this terminology without explaining its meaning for your purposes. – GEdgar Apr 21 '15 at 18:19

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.

• @JohnDoe: Thanks a lot for granting the bounty and even more pleasing (due to the fundamental nature of this question and the many different answers): Thanks for accepting my answer. – Markus Scheuer Apr 26 '15 at 19:45

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.

• Supposing the second criterion (I wonder about that myself), despite same limit convergence, or lack thereof, although the series may be called equal, but surely they're not the same, right? Also, suppose two series diverge to $+\infty$ --- again bunching them as the same doesn't seem right. So the question also comes down to what is meant by same. I take no issue with your reply (I have +1 it, in fact) --- the above are just my own musings. – prime4567 Apr 22 '15 at 16:32
• @prime4567 I agree, we can reduce the problem to a formal definition of the word "same". And for the series that diverge to $+\infty$, we can, for instance, have the notion of "rate of divergence" (basically, big-O notation), and call the series with the same rate as "more same than others"=). – TZakrevskiy Apr 22 '15 at 16:45

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".

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.

• Thank you for the answer. It was helpful to think about the well-ordered sets. – John Doe Apr 26 '15 at 19:05

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!

• I'm flabbergasted that this pseudo-mathematical argument pops up every time again. It is clear that the sum of all naturals does not exist. Period. – Han de Bruijn Apr 21 '15 at 18:47
• "The sum of all naturals", like all mathematical constructions, is whatever we define it to be. – Sam Clearman Apr 22 '15 at 16:37

The best way to think about your question is that you are considering the set of all sequences $(a_1,a_2,\dotsc)$. A priori, two sequences are equal if and only if each of their terms is equal.

However, we can define various equivalence relations on sequences. In particular, if we are considering the sequences as the terms of a series, we might want to define the relation $(a_n) \sim (b_n) \iff \sum a_n = \sum b_n$, given some notion of summation.

In this case it would be advisable to either restrict our attention to the summable sequences, or consider all divergent series to be distinct from one another, as a summation technique typically does not have much to say about a sequence that doesn't sum. However we certainly could define $(c_n) \sim (d_n)$ for all divergent $c_n, d_n$ if we really wanted to.

We might also want to consider "finer" equivalence relations that distinguish between some sequences that might have the same sum. (By "finer", I mean a relation $\sim'$ s.t. $x \sim' y \implies x \sim y$ - any relation that does not have this property, we probably don't want to consider.) For example we could define a relation $\sim'$ by $(a_i) \sim' (0,\dotsc,0,a_1,a_2,\dotsc)$ for all $(a_i)$ and all numbers of leading zeroes. As has been noted, this relation is not compatible with all types of summation, but it works with the ordinary kind. Another similar example is that we could define $(a_i) \sim' (b_i)$ if $(a_i) = (b_{w(i)})$ for some bijection $w: Z \rightarrow Z$. This of course is not compatible with ordinary summation, but it is compatible on the subset of absolutely convergent series.

It is interesting to note that one can in fact define the real numbers as equivalence classes of Cauchy sequences under a certain equivalence relation. In our terms this amounts to only considering summable sequences (using ordinary summation) and defining sequences to be equivalent if they have the same sum - the point is that these definitions can be made without reference to the existence of sums, or real numbers. For more information see http://en.wikipedia.org/wiki/Construction_of_the_real_numbers#Construction_from_Cauchy_sequences

There is also a whole theory of metrics on sequences which is analogous to $L^p$ spaces of functions, for more information see: http://en.wikipedia.org/wiki/Sequence_space