# Are there differing definitions for the value of an infinite series?

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 en.wikipedia.org/wiki/Divergent_series . –  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

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.