Thanks Plato.

As always, you show me where I misunderstood and you provide helpful guidance. These video courses aren't easy when it's impossible to ask the professor questions.

This particular lesson is titled "Sequences and Series" and introduces Limits. The chapter does not go into "converging" nor "diverging" limits. It does describe sigma, its notation and what it means in general terms. It is not much more enlightening than the discussion of asymptotes in the conics chapter. But it does describe limits in the context of infinite series.

The distinction you make does make sense to me.

I wish I could post the part of the video in question but it's proprietary.

In the part in question, the Professor shows, on the screen:

$\displaystyle

x=\dfrac{1−r^{n+1}}{1−r} \ \ r<|1|\ \ \sum_{i=0}^{\infty}r^i=1+r+r^2+r^3+...=\dfrac{1}{1-r}$

And, verbatim, he says, "Now, imagine the following: Imagine the r is less than one and greater than minus one. In other words, the absolute value of r is one. That means that r to a large power starts getting smaller and smaller, doesn't it? Like, if r were a half, then one-half to a billionth power is really small. So, in the **limit** *[emphasis mine]*, as we go to infinity, that r to the n plus 1 begins to disappear. And, in fact, if you add, from i equals zero to infinity, r to the i; that's one, plus r, plus r squared, plus r cubed, forever, the r to the n plus one disappears and your formula is one over one plus r. And, of course, you have to make sure the absolute value of r is less than one. If r is greater than one or equal to one in absolute value, **the series doesn't add up to a finite number** *[emphasis mine]*."

He goes on the describe Xeno's Paradox (with which I was already familiar) and concludes his description with, "...he argued you would never get across the room.And you can sort of see this as an infinite series."

The series appears on the screen: $\displaystyle \sum_{i=0}^{\infty}(\dfrac{1}{2})^i = 1 + \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} +... =? $

And he continues, "We're adding up one as the first step, and then a half, and then a quarter, and then an eighth, et cetera. But we know better now. Don't we? Now that we've studied these infinite series, we're adding this up to infinity. And, of course, if you add up to infinity, it's one over two, one-half to the i, from i equals zero to infinity. According to our formula, it's one over one minus r. Which is one over one minus one-half. And that equals 2. So, this infinite series, out to infinity, adds up to two. **Xeno does cross that two-meter walk**."

But in truth, if I understand you correctly, Xeno is still correct. The closest you can approach that limit of 2 meters is just up to, but never quite at the 2-meter limit.

The professor uses the same logic to prove that 0.9999999999999999...$\infty\ \ =\ \ 1$ And he adds, "Wow, the sum of nine-tenths plus nine-one-hundredths, and so on to infinty, is equal to 1."

To my mind, this last statement is false. The limit of this decimal, consisting of an infinite number of 9's, does approach 1, but is never exactly equal to it.

Should it be assumed that any summation of an infinite number of items is always the limit that the sum approaches, even though it doesn't say "limit"? Is that the part I'm missing?

The significance of limits and their calculation isn't lost on me. But I need to make sure I comprehend the concept.

This is really long but I appreciate all your help.