I want to know more about series convergence (elementary)

[1MileCrash]

I am able to use a variety of methods to check to see if a series converges, and I can do it well. However, it's not something I feel like I've intuitively conquered.

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?)

 

[micromass]

I think I understand your concern. But I'm afraid there is no easy answer. You find it not intuitively true that the series 1/n diverges but 1/n² does not. I don't think I can explain to you why, except to say that they do.

However, I can give you an intuitive proof why 1/n converges:

$$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8} +\frac{1}{9}+\frac{1}{10}+...$$

$$\geq 1+\frac{1}{2}+\frac{1}{4}+\frac{1}{4}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8} +\frac{1}{16}+\frac{1}{16}+...$$

$$\geq 1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + ...$$

So this shows why our series diverges. We can even use this to find how fast the series diverges.

Likewise, one can indeed show that 1/x has infinite area and 1/x² has finite area.

I know it isn't intuitive, but it's something you need to get used to.

 

[chiro]

Have you come across formal methods for convergence involving norms or are you in the early stages of your degree?