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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.2011-11-16
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    (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?2011-11-16
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    Convergence is not an issue here. The question of "identity" arises already when you write $2+3=5$.2011-11-16
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    @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: http://en.wikipedia.org/wiki/Eval#Lisp2011-12-17

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