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I don't understand why the series 1/x diverges. I mean, I do, in that I know the integral test will give me the limit as x -> infinity of ln|x| which grows without bound, and I understand why the integral test makes sense, but I don't

*get it*.

Why does it matter how quickly the function approaches 0 on an infinite plane?

Is there really an infinite area under the curve 1/x, but a finite one under 1/x^2? Why? What's so different about the two?

Does someone understand my concern? Is there some link between 1/x being the "standard" for whether or not a series converges and the behavior of ln x (increases extremely slowly?)