[0xff00ffee]

Yep. Agree 100%. It is like the blue dress.

I think the problem is the repeating function. Infinite things are non-intuitive and should be presented differently.

Even here on HN you still see people confused about "convergence" and "identity". 0.999... doesn't CONVERGE, it literally is 1.

I suspect this persists even with students that have had second year college calculus that discusses convergent series and sums.

[tom-thistime]

Fine. Define 0.999... as the limit of the series sum(n=1 ... N)(10^-n), as N-> infinity. This is standard high school calculus. "Number" and "series" and "limit" and "convergence" don't all mean the same thing. However this number is defined as the limit of a convergent series. So the question really is meaningful. (One clue that this question is meaningful is the amount of space introductory calculus textbooks use to address it.)

Because I can still ask, in black and white, what law of "equality" do I use to establish that my limit equals 1? (It does, if I import the definition of "equality" from the real numbers. That's what they do in calculus class. )

[tom-thistime]

Thanks to a commenter who pointed out that my sum above should be

sum(n=1 ... N)(9*10^-n).

I can't, uh ... fully endorse that comment, which is not entirely accurate and doesn't answer my question. But I sure did miss the '9'.

[0xff00ffee]

I'm not sure if you are asking for an answer or a rhetorical question? I'll assume the former.

Your terms are bit jumbled, so let's keep it simple: you're asking how to prove if an infinite sum converges and what its value is. Convergence proofs require analytic thought: meaning there may not be an immediate look-up. You need to convert the problem into the known corpus of convergent sums or use one of many tests (bounds test, integral test, etc) to show it converges analytically. Which you only learn through experience and memorization (unless you want to re-prove hundreds of series... maybe you do!) Fortunately this one is easily re-written as a known convergent sum.

First, you missed a term in your sum (9), re-written here:

sum(n=1..inf) 9 * 10^-n

Step 1: you pull out the 9 and it becomes 1/10+1/100+1/1000...

Step 2: Then we shift to n=0 by subtracting 1/10^0 from the series so that it is in the form n=0..k-1

1/10^0 + 1/10 + 1/100 + 1/1000 + ... + 1/10^-n - 1/10^0

Step 3: Now we've got ourselves a geometric series of just 1/10^n .. wikipedia does a great job explaining the sum convergence for GS from n=0...inf: https://en.wikipedia.org/wiki/Geometric_series

Step 4: compute geometric convergence

(1-r^n)/(1-r) = (1-(1/10)^n)/(1-1/10) = 1/(1-1/10) = 10/9

So we have 10/9 as the solution to Sum[n=0...inf](1/10^n)

Step 5: the remaining arithmetic

Now subtract our 1/10^0 ... and then * 9 = 1