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Gabriel's Horn & Infinite Series via Limits

In my AP Calculus BC class we have just gone through improper integrals. I understand how to get the determinate answer and when it is indeterminate, etc. but conceptually I don't actually understand how the integral of 1/x2 from 1 to ∞ can be 1 (or even finite, for that matter). I know about infinite series (that is our next unit) and I have to say, infinite things in general confuse me.

I can understand that some infinite series sums up to 1 (the infinite rectangles summing up the area under the curve), but that's as far as my understanding goes. I don't understand why it stops at 1. I know, from a limit perspective, it approaches 1; like I said I understand all the math, it's just conceptually confusing.

It's not interfering with my ability to solve problems, it just is something I can't find an answer to. Sure, the numbers get smaller and smaller until they amount to nothing, but that only applies to limits - I can't understand anything about infinity without limits.
I can't understand anything about infinity without limits
That's good, because in calculus all infinities come with limits.

There is nothing particularly complex about a plain converging sequence. The idea is that you can find a tail which is as close to some point as you'd like. Take a sheet of paper, choose a limit point C, then put a point anywhere on the sheet at most 1 cm away from C, that's A1. Then put a point anywhere on the sheet at most 1/2 cm away from C, that's A2. If you continue, you'll make a sequence {An} that converges to C. In fact, if you instead put multiple points at each distance level, you could make all sequences that converge to C that way. Of course, in calculus, you work in a line, rather than a plane. That only makes thing simpler though.

A series converges iff the sequence of its partial sums converges. You can construct any series by taking some sequence and summing over Ai-Ai-1. It's only the presence of summation operator that makes series harder to work with than plain sequences.

An integral converges if you can give lower and upper approximations to the area under a curve that are as close to each other as you'd like. Riemann made his approximations by drawing thin rectangles, so you will too. While it could be reduced to a sequence, a proper integral is fairly nice so you don't have to actually look at it as a limit of two sequences.

An impropper integral is a limit of proper integrals that approach the "bad" point. The integral of 1/x2 converges because the sum of 1/n2 i.e. a bunch of rectangles 1 unit wide and 1/n2 units high, converges (though it's more often explained the other way).
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Someone else managed to explain it to me in a way I understood.

I'll take your point on paper example. If I keep drawing another point between the last point and the actual point, using a mathematical formula or such, then there is no way to overshoot the actual point because of how there is division involved. If I think about it this way, it makes sense.

My problem was that I thought the distance between all the points added up to more than the distance between the first point you placed and the actual point, because of the addition of infinite numbers, no matter how small. But if I think about this using the same logic as placing the points, I see how the distance can't overshoot the real distance.

The most important advice they gave me was that I'm not supposed to be able to understand this without limits anyway, because it's all just theory. I guess that what I really needed to understand was that one way of thinking about it works and another doesn't, like how template instantiation failure is not an error.

Thanks for trying to explain it to me, though - you didn't actually help me understand, but your point on paper let me explain my understanding.
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