4
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I would like to express

$$1+22+333+4444+\cdots$$

using $\Sigma$ notation, and have no clue where to start.

After $999999999$, comes 10 $0$s, then 11 $1$s.

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    $\begingroup$ Well, it appears useful to note that a string of, say, k ones can conveniently be expressed as $\frac{10^k-1}{9}$ . Similarly, a string of k i's (for any non-zero digit i) is just $\frac{i(10^k-1)}{9}$. $\endgroup$ – lulu Jul 7 '15 at 1:37
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    $\begingroup$ It can be concisely expressed using the symbol $\infty$. $\endgroup$ – dalastboss Jul 7 '15 at 1:42
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    $\begingroup$ @dalastboss That is not what the question is asking. An expression has a sense and a value. The expressions value is $\infty$, but the question is not what the value is but how to express the sense. If you don't distinguish between these two notions, you are in the uncomfortable position of saying (for example) that the statement $e^{i\pi} = -1$ is trivial, since of course the two sides have the same value. What makes this and similar statements interesting is that the two sides have different senses. This distinction goes back to Gottlob Frege. $\endgroup$ – MJD Jul 7 '15 at 12:02
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    $\begingroup$ @MJD I believe that was intended as something of a joke, and a rather funny one at that. $\endgroup$ – Jared Smith Jul 7 '15 at 14:33
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The terms of the sequence are

$$a_n =\left(n-10\cdot\left\lfloor\frac{n}{10}\right\rfloor\right)\cdot\frac{10^{n}-1}{9}$$

the sum of your series is $\infty$

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$$\sum_{n=1}^{\infty}\left(n-10\cdot\left\lfloor\frac{n}{10}\right\rfloor\right)\cdot\frac{10^{n}-1}{9}$$

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Considering $$ 4444 = 4*10^{3} + 4*10^{2} + 4*10^{1} + 4*10^{0} $$

I think a double sum and modulo is a lot more intuitive: $$ \sum_{n=1}^{\infty}\sum_{m=1}^{n}(n \text{ mod } 10)*10^{m-1} $$

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