They both get closer to but never reaching zero. Is the reason simply that one gets 1/x^2 gets closer to zero faster? So whats the threshold for something to be considered finite or infinite?
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>whats the threshold for something to be considered finite or infinite?
It’s not a matter of opinion, things either are finite or they are not.
If you are asking about the integral from 1 to N of these functions as N tends to infinity, then the answer is, at an ELI5 level, just “because that’s how the functions behave”.
* The integral of 1/x is ln(x), and the limit of ln(N) as N tends to infinity is undefined – i.e. the integral is not finite.
* On the other hand, the integral of 1/x^(2) is -1/x, and the limit of 1/N as N tends to infinity is zero. This means that the integral from 1 to N converges to 1.
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