I made drawings of both functions.
With respect to 1/x^2 I was very strict: I calculated the area under the straight line connecting ( 1,1) and (2,1/4) and so on. This gives two series that are both convergent by comparison with the series 1, 1/2, 1/4 ....The area of this series is larger than the area we are interested in, so our area under the integral is finite, as expected.
With respect to 1/x I was very lenient, and I calculated the area connecting (1,1/2) and (2,1/2), (2,1/3) and (3, 1/3) and so on, which gives the harmonic series of which we know that it diverges. The area that we calculated is smaller than the area we are interested in, so the area of the integral here is infinite.Harmonic series was already suggested by Cardinal, thanks for that suggestion.
So even when you are very strict on the first, and very lenient on the second function, the first is clearly finite, and the other infinite.But in the end you use simple geometry- pre calculus I suppose- and combine it with knowledge with respect to convergence of series.
I now realize that Mark Viola hinted this already in his post "Frank just curious". His mysterious response was alluding to what would happen if you inspect the areas, thereby getting these series he made reference to. I guess he wanted to just give a hint;)!
It is common knowledge that both functions blow up at x=0. The function y=1/x diverges also at the tail, so I will leave it at that. By inspecting the areas of y=1/x^2 one gets a series that keeps adding terms, so the area is infinite if one includes x=0, as expected.
I took it from x=1 to make calculations easier and because I thought the question was with respect to the tail of both functions.Since it is about the tail , choosing the left limit of the integration seems arbitrary.
My response is late: a student at the calculus course I am mentoring asked the question, now being about x going from 1 to infinity, I saw the page of this site, and I didn't notice any posts that did the same thing as I did. It was very simple, but I thought that this was meant by "geometrical" in the original question. I posted because I liked to have some feedback on the content, not because I wanted a medal;)!
As to your comment hardmath: I thought it was clearly understood that the question was not about the blow up of both functions.
Are there follow up answers that I didn't see, or did the discussion pass away some four years ago? If the last is true, is it senseless to add a post after a long time?
It seemed to me that the original post actually wanted to point out that things were questionable because they were not intuitive, brought philosophy into the discussion, and after that I saw some reference to space time. After evaluating the level of some of these comments- only a few of them- I thought that simple calculations would at least meet that level;)!
Ad van Straeten
In fact this was a very simple answer to a very simple question. Both are not consistent with the advanced level of the other discussions at this site. But it is still an answer that was not provided yet.