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Previously, I'd been taught that convergent series "converge" to a certain number. Taking a convergent improper integral as an example, the value of the integral would be "definite" and not infinity.

However, every time I think about this, it doesn't make sense to me (I suspect it's probably because of a flaw in some basic understanding).

Anyway, my thinking is that since we are adding infinitely many values (as we do with convergent series), the sum of infinitely many values would be infinite. However, according to the accepted answer for this question,

[That is] incorrect. As long as the positive values an you are summing decrease to 0 fast enough, the sum... will be finite.

While I don't doubt this, it still doesn't give me any intuition. My thinking is still that, regardless of how fast you approach zero, since you have "infinite time" to sum "infinite numbers", your speed at which you approach zero wouldn't matter. The analogy doesn't really convince me. Could someone help clear up (conceptually) whatever misunderstanding I may be having?

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    $\begingroup$ Take the geometric series 1+half+quarter and so on. As you said, it's always getting bigger. However, it can never reach 2, always going half the distance left to it. Saying it equals 2 has always felt a little sloppy to me, because convergence means that you can get as close as you want to 2 in the limit as the sum increases without bound, not that it will "be" 2. But, I'm no expert on that. $\endgroup$ – Benjamin Thoburn Dec 1 '19 at 16:16
  • $\begingroup$ I believe the common definition of convergence is partial sums monotonic (I.e. always increasing or decreasing) and bounded (doesn't pass a certain point) for these kinds of non-alternating-type series. $\endgroup$ – Benjamin Thoburn Dec 1 '19 at 16:22
  • $\begingroup$ Perhaps of interest. google.com/amp/s/www.livescience.com/amp/… $\endgroup$ – Peter Szilas Dec 1 '19 at 16:29
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I think this is the root of your problem.

since you have "infinite time" to sum "infinite numbers",

It took mathematicians many centuries (from Zeno to the rigorization of analysis) to learn how to deal with infinity. Along the way some ideas that seem intuitive had to be discarded.

When you sum an infinite series you are not actually adding together infinitely many numbers. The only additions you ever perform are finite sums. Moreover, whatever adding you do, you don't do one after another in real time.

Instead you look ("all at once") at the sequence of all (finite) partial sums. If that sequence has a limit, you define that limit as the sum of the series.

Note that asserting that the sequence has limit $L$ is not saying it "gets closer and closer to $L$". There is no "closer and closer", which again suggests the passage of time. The limit is $L$ if the partial sums are (not get) as close to $L$ as you might wish as long as you add up sufficiently many terms. Never all the terms.

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The key is partial sums.

Consider the series

$$1+\frac12+\frac1{2^2}+\frac1{2^3}+\cdots$$

It is an easy matter to prove that all partial sums ($1,\frac32,\frac74,\frac{15}{8},\cdots$) are smaller than $2$, regardless the number of terms.

$$\Sigma_{n}<2\implies \Sigma_{n+1}=\frac{\Sigma_n}2+1<2.$$

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A simple but effective way to see how the sum of infinitely many terms can converge to a finite number is the following geometric series

$$\sum_{n=1}^\infty \left(\frac14\right)^n$$

and its geometric visualization as a infinite sum of squares, from which we can see that the value for the series must be less than a finite quantity since it is contained into a unitary square

enter image description here

(credits)

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    $\begingroup$ Looks more like an illustration of $\sum_{n=1}^\infty\frac1{2^{2n}}$ to me. $\endgroup$ – Lord Shark the Unknown Dec 1 '19 at 16:19
  • $\begingroup$ @LordSharktheUnknown Yes of course! I fix that. Thanks $\endgroup$ – user Dec 1 '19 at 16:21
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Say Zeno and a tortoise are running a 2 kilometre race. Zeno gives the tortoise a 1 kilometre head start. Zeno runs twice as fast as the tortoise. When Zeno reaches the point that the tortoise was at the start, the tortoise has run another 500 metres (1/2 kilometres) reducing the gap to 500 metres. The gap keeps on reducing until the end of the race, when the gap is zero (the tortoise runs half the distance in the same time as it is half as fast as Zeno). The sum of all of the gaps (in kilometres) is 1+1/2+1/4+1/8+... and that must equal 2 as the race is 2 kilometres long.

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