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?
In: 59
Because of how these integrals are *defined*.
First off, be careful. If you start integrating at 0, they’re both infinite. If you start at 1, only one is.
The integral from 1 to infinity is a limit : you integrate from 1 to x, this defines a function F(x) -the antiderivative if defined- and if F(1)-F(x) has a finite limit as x goes to infinity then that value becomes *by definition* the integral from 1 to infinity.
With 1/x, the antiderivative is the logarithm, which grows infinite with x, so no finite limit. With 1/x^2, you obtain a finite limit, hence the value of the integral.
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