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Inspired by a fierce online debate over the question whether or not $1 = 0.\bar 9$ someone claimed that there are mathematicians who do not accept the definition of the value of a (convergent) infinite series $$\sum_{n=0}^\infty a_n$$ as the limit of the sequence $A_N$ of partial sums, $$A_N = \sum_{n=0}^N a_n$$

Is there any merit to that claim?

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Yes, but every summation method I know gives this answer. See . – Qiaochu Yuan Oct 5 '11 at 3:52
The question whether $0.999\ldots=1.000\ldots$ should not be resolved by summing an infinite series. Use the following argument instead: If $0.999\ldots<1.000\ldots$ there would have to be some numbers in between the two, but there is no decimal representation possible for such numbers. The only way out of this dilemma is declaring $0.999\ldots$ and $1.000\ldots$ equal, i.e., infinite decimal representants of the same number $1$. – Christian Blatter Oct 5 '11 at 12:13
up vote 6 down vote accepted

The phrasing of the statement is misleading. It's not that there are mathematicians who do not "accept the definition", rather it's that all mathematicians are aware that this is just one of several useful definitions of the sum of an infinite series, yet all the other alternative definitions (e.g. Cesàro summation) agree with this one when it exists (namely when the set of partial sums actually has a finite limit).

So in other words, yes there are alternative summation methods, but no there is no "rejection" of the standard one for series in which it exists.

Of course one can go ahead and define the "sum" of an infinite series to be whatever one wants (you can define $\sum a_n$ to be $a_1$, or $17a_3+a_4^2$, or 9, or whatever), but it's unlikely that such a definition which doesn't agree with the ordinary one would be useful for anything. The expression $0.999\bar{9}...$ can also be "defined" to be 7 or whatever but that doesn't change the fact that it has a certain specific meaning in mathematics, and that meaning makes its value be exactly 1. Someone who denies that would have a hard time earning the title "mathematician" by other mathematicians.

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