# How equal are a series and its sum?

This is a question about the logic of mathematical language concerning infinite series.

It's normal to say $\sum\limits_{n=0}^\infty 2^{-n}=2$. This type of equation is often given in the form of a notational introduction within the definition of "converge(nt)". But it's also normal to assert things about $\sum\limits_{n=0}^\infty 2^{-n}$ and deny the corresponding statements about $2$:

1. $\sum\limits_{n=0}^\infty 2^{-n}$ converges, but $2$ does not "converge".
2. $\sum\limits_{n=0}^\infty 2^{-n}$ is an infinite series, but $2$ is not.

On the surface, this looks like a violation of the basic substitutability of equals for equals. I see two possible explanations:

First, the problem result from talking about $\sum\limits_{n=0}^\infty 2^{-n}$ but implicitly referring to its form. For example, in "6/3 is a fraction but 2 is not", the idea "is a fraction" refers not to the value of 6/3 but to its form. This seems plausible and attractive, especially for saying "is an infinite series", but it seems to be a stretch for "converges". For example, to say that a nested sum $\sum\limits_{n=0}^\infty \ \sum\limits_{m=0}^\infty \ldots$ "converges" (treating it as a sum on $n$), we require that the inner sums be evaluated. This does not feel like a description of form alone.

Second, the problem might occur because the equality of $\sum\limits_{n=0}^\infty 2^{-n}=2$ is not in fact sincere equality: It means something other than logical identity. This interpretation is strongly favored by the fact that it appears in a definition! (Presumably we would not be entitled to redefine logical identity.) Thus there is no reason to expect substitutability, and there is no problem. But this seems disingenuous: In many contexts, we freely substitute series and their sums. We also use this "$=$" symbol symmetrically and transitively, mixing it without comment with normal equality.

Have I correctly understood normal usage? Is either of these interpretations the "correct" one? Is there a "logician's solution"?

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"is not in fact sincere equality" - right, "=" in $\sum\limits_{n=0}^\infty 2^{-n}=2$ has a special meaning. The situation is the same as with limits in calculus. – J. M. Nov 16 '11 at 1:37
(1) I don't understand your example you believe suggests series convergence isn't a description of form. Elaborate? (2) In math, equality of expressions is not the same as identicalness of expressions. Surprised? Why? – anon Nov 16 '11 at 1:43
Convergence is not an issue here. The question of "identity" arises already when you write $2+3=5$. – Christian Blatter Nov 16 '11 at 8:16
@Christian Interestingly, this doesn't seem confusing to someone who has a bit of computer science background. '2+3' and '5' are different expressions but they both have the same value (5) when evaluated. See: en.wikipedia.org/wiki/Eval#Lisp – Chris Taylor Dec 17 '11 at 12:59

I think you run into these problems long before you hit infinite series. There are statements that are true about $2+2$ but false about $4$. If you can understand the extent to which substituting equals for equals works for $2+2=4$, where you don't have the distractions of infinity, you may be well on your way to answers to your questions.

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In the sense of the tag (logic), there is no logic involved. Rather one can keep in mind that, loosely speaking, a series is really the sequence of its partial sums.

Somewhat more rigorously, start from a sequence $(a_n)_{n\geqslant0}$, say of real numbers to keep it simple. Then the series associated to $(a_n)_{n\geqslant0}$, often denoted $\sum\limits_na_n$ is the sequence $(A_n)_{n\geqslant0}$ defined by $A_n=\sum\limits_{k=0}^na_k$ for every $n\geqslant0$. The sum of the series $\sum\limits_{n}a_n$, often denoted $\sum\limits_{n=0}^{+\infty}a_n$ when it exists, is the real number defined as the limit $\lim\limits_{n\to\infty}A_n$ of the sequence $(A_n)_{n\geqslant0}$.

One sees that the assertion $\sum\limits_{n=0}^{+\infty}2^{-n}=2$ is in fact a shorthand for two successive statements: first, based on $a_n=2^{-n}$ for every $n\geqslant0$, the sequence $(A_n)_{n\geqslant0}$ defined above converges; second $\lim\limits_{n\to\infty}A_n=2$.

Of course, confusion may occur as soon as one uses $\sum\limits_{n=0}^{+\infty}a_n$ to denote $(A_n)_{n\geqslant0}$, but otherwise everything works fine. As an example, note that $\sum\limits_{n=0}^{+\infty}2^{-n+1}=\sum\limits_{n=0}^{+\infty}(3/4)^{n}$ since both these real numbers are the number $4$, while $\sum\limits_n2^{-n+1}\ne\sum\limits_n(3/4)^{n}$ since there exists at least one integer $n\geqslant0$ such that $2^{-n+1}\ne(3/4)^{n}$.

Edit A good point mentioned by @Gerry is that the same distinction should be kept between finite sequences and their sums. The sequences $(a_n)_{0\leqslant n\leqslant3}$ and $(b_n)_{0\leqslant n\leqslant3}$ defined by $a_0=a_1=a_2=a_3=3$ and by $b_0=b_1=3$, $b_2=4$, $b_3=2$, are not equal because $a_2\ne b_2$, for example, although $a_0+a_1+a_2+a_3=12$ and $b_0+b_1+b_2+b_3=12$ are equal.

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When we write "2+3=5" then we don't mean that the heap of pixels "2+3" is equal to the heap "5" but that we have an abstract notion of the objects $2$, $3$, $5$ and furthermore a function ${\rm plus}$ that assigns to any given pair of numbers $(x,y)$ a value ${\rm plus}(x,y)$ written as $x+y$. Then the equation $2+3=5$ expresses the fact that the value taken by ${\rm plus}$ on the pair $(2,3)$ is $5$.

Similarly, given an infinite sequence $a:=(a_0, a_1, a_2,\ldots)$ of numbers there is a (logically quite involved) function ${\sum}$ which assigns to this sequence either the value ${\tt undefined}$ or a certain number. An equation of the form $\sum a= \sigma$, where $\sigma$ is a certain number, is the statement that the value assigned by $\sum$ to the sequence $a$ is $\sigma$.

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There is very often a difference between the meaning of the same phrase or expression according to the context. Writing an infinite summation in the context of a larger expression means that you implicitly assert its convergence, and use the limit value in the expression. This is similar for a direct sum of subspaces (in the aspect that there is an implicit claim as a side product of a value produced). If you write an infinite summation by itself (talking about its terms, its convergence) you are probably speaking about its form. Similarly if you say some vector is a linear combination of some others, the linear combination means its value, while if you say a linear combination is nontrivial, you are referring to its form ("a family of vectors is independent if no nontrivial linear combination of them equals the zero vector").

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