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So, I'm sure this has been thought of and said before but I'm curious.

So $\sum \frac{1}{2^n}$ can be thought of by filling up a square. First we color in a whole square and then we draw a second square and fill in half. Then fill in half of what's remaining, that is one quarter of the square. Then fill in half of what's remaining, that is one eighth of the square. And so on. In this way we can see the series converges to 2.

Now, suppose we have a convergent series. Then we can scale said series so it converges to 1. So, similar to above, we should be able to shade in a square in a non-overlapping fashion (though likely not nearly as nice).
So, we can say a series is convergent exactly if we can color in a square in a non-overlapping manner corresponding to the individual terms of the series. Do you folks think this is a helpful way to explain series to students?

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  • $\begingroup$ Yes, thanks. Sorry I typed on my phone $\endgroup$ – Aaron Zolotor Apr 16 at 3:40
  • $\begingroup$ Sure, check this question out ... math.stackexchange.com/questions/3177728/… $\endgroup$ – mjw Apr 16 at 3:46
  • $\begingroup$ I'm not sure it is so helpful as the coloring will in general not be naturally defined; for instance, how would you describe the coloring coloring corresponding to $\sum_{i=1}^\infty (i(i+1)\dots (i+k))^{-1}$? $\endgroup$ – TomGrubb Apr 16 at 3:47
  • $\begingroup$ If the students are not familiar with the logical foundations of $\Bbb R$, they will have questions and debates about whether the series sums to $2$ or to something "infinitely close to but less than $2$". $\endgroup$ – DanielWainfleet Apr 16 at 9:49
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In a strictly intuitive sense, the answer would be yes. You would probably have to develop some conventions for, say, "negative" squares and such - maybe draw them in a different color perhaps. And we would need a notion for reductions in the amount colored, which I guess could be analogized as erasing in some sense.

If $\sum_n a_n$ converges to $S$, then in such a manner we would be able to somehow color in $S$ unit squares. If the sum does not converge, then at some point we're either oscillating somewhere (so we're basically constantly erasing and redrawing squares without getting anywhere), or the sum grows without bound (and we're basically drawing infinitely many squares).

Of course there are issues aside from those in my first paragraph. As keenly noted in the comments, not every summation would lend itself to such a nice method. Geometric sums, like $\sum_n 1/k^n$ for $k$ a fixed constant certainly do, but what of $\sum_n 1/n$? Or $\sum_n 1/n!$? Or one of the weird and wacky summation formulas for $\pi$?

The idea of coloring in squares is a nice way to build intuition about convergence and provides a nice tool for "proofs without words" for such sums, but sadly its utility in terms of problem-solving and proof-writing dies out pretty fast when we consider anything that isn't particularly simple. (Not to undermine intuition of course: as a teaching tool it's absolutely important. It's just also worthwhile to recognize its limitations, and the limitations of what tools you use to impart that intuition.)

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