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I recently learned that an infinite series is just the limit as the partial sums of a sequence approach infinity. This idea is expressed as:

$$\sum_{k=0}^\infty a_k \stackrel {def} = \lim_{n \to \infty}s_n$$

However, I intuitively feel that there are some times that an infinite summation appears more "naturally" in math. Namely, by long division, one can observe that $.\overline{3}= \frac13$. Wouldn't this show (or prove) that $$\frac {3}{10}+\frac {3}{100}+\frac {3}{1000}+...=\frac13$$ regardless of the limit definition of an infinite summation? Or, as another example, isn't it true that $$3+\frac {1}{10}+\frac {4}{100}+\frac {1}{1000}+...=\pi$$ To me, the definition of a series seems to define something that already exists in math. I suppose my question is: do you need an infinite series/sequence to define a non-terminating decimal? Are we defining something that is already defined? I'm confused by the train of logic used here.

The only thing I can think of is that long division resulting in a non-terminating decimal isn't defined until you define sequences and series. But from what I can gather, it seems like you can define decimal expansions without series, and that confuses me. Any clarity would be appreciated!

Bendy2021
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    These sums that appear as natural are why we define them precisely. Applications motivate abstractions. – John Douma Jan 26 '22 at 01:50
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    What does it mean, exactly, to say that $.\bar 3=1/3$? You could try to give it a definition directly in terms of long division, but I don't think that's what you really mean by this equation. Think it over. – Kevin Carlson Jan 26 '22 at 01:57
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    As others have said, when you carry out the long division, are you doing an infinite number of steps? Or are you recognizing a pattern and making a claim about what happens as the number of steps tends to infinity? That sounds like a limit to me... – C Bagshaw Jan 26 '22 at 02:00
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    you might be interested in this one: https://math.stackexchange.com/questions/2073041 – 311411 Jan 26 '22 at 03:09
  • I think I'm gathering that it would be safe to say that the (or at least a) definition of a non-terminating decimal is a sequence of partial sums as n approaches infinity. – Bendy2021 Jan 26 '22 at 04:01
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    25 centuries ago in Greece they would have demanded that you explain in specific and complete detail just exactly what the "$...$" means in $3/10+3/100+3/1000+...$. They would say they know what a sum of finitely many numbers is, but they would demand that you explain what a sum of infinitely many numbers is, how it is defined and how you know it exists. These demands were thoroughly met by others, a little later, only in the 19th century AD. – DanielWainfleet Jan 26 '22 at 10:22
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    $\Bbb R$ is defined as an ordered-field extension of $\Bbb Q$ in which every non-empty subset with an upper bound in $\Bbb Q$ has a $\lub$ in $\Bbb R.$ So you can define $0.\bar 3$ as the $\lub$ of the set of all finite partial sums, which in $\Bbb R$ is $1/3. $ You cannot do this a priori without defining $\Bbb R$. We can extend $\Bbb R$ to larger arithmetic structures (that are also ordered fields) that do not have that $\lub$ property. – DanielWainfleet Jan 26 '22 at 10:43

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