What makes one integral converge and a similar integral diverge, e.g., $\int\limits_1^\infty\frac1x dx$ vs $\int\limits_1^\infty \frac1{x^2}dx$? The area under $\dfrac 1x$ curve within $[1, \infty)$ is considered to be infinite but for $\dfrac 1{x^2}$ curve, it is $1$.
Can someone please explain this to me intuitively?
 A: As many have pointed out in the comments, $\frac{1}{x^2}$ gets smaller a lot faster than $\frac{1}{x}$. As $x$ gets large the comparison between the two fractions is astronomical.
Consider $x=10^{12}$ (a trillion). $\frac{1}{x}$ is one trillionth but $\frac{1}{x^2}$ is a trillionth of that. What does that difference look like? Suppose you are standing at $x=10^{12}$ and the curve $y=\frac{1}{x^2}$ looks like it is one foot above you. Then The curve $y=\frac{1}{x}$ appears to be a trillion feet above you.
A trillion feet is about $189$ million miles which about twice the distance as the sun is from the earth. Can you see how the two curves are really not that similar in the way they tail off?
A: Although "intuition" varies from person to person, it would be accurate to note that $\int_1^\infty {1\over x^p}dx$ is (by the fundamental theorem of calculus) easily verified to converge for $p>1$, and to diverge for $p\le 1$.
For me, the "intuition" about this comes entirely from that computation. I have no genuine physical intuition for those infinite integrals. But I know the fundamental theorem of calculus, and can do the pursuant easy algebra, so I know what is true.
And I've known what is true for many decades, so it is "part of my intuition".
I seriously think that is possibly the most accurate-and-honest answer you can get on this.
