Geometric (or Intuitive) proof of the improper integration of $\frac1x $

Question

From a mathematical standpoint, I understand and I can solve the following: $$ \lim_{M\rightarrow\infty} \int_1^M \left({1 \over x}\right) \rightarrow \infty $$ Additionally, $$ \lim_{M\rightarrow\infty} \int_1^M \left({1 \over x^2}\right) \rightarrow 1 $$

This all makes mathematical sense to me. It's the geometric parts that confuse me. The family of of $ 1/x^p $ graphs look very similar to me, so it makes me wonder why $ 1/x $ doesn't converge to some value as well.

Especially considering the fact when you rotate $ 1/x $ and calculate the volume of that shape; it converges to some value. Again, mathematically, this makes sense because:

$$ \lim_{M\rightarrow\infty}\int_1^M \left({1 \over x}\right) dx > \lim_{M\rightarrow\infty} \int_1^M \pi\left({1 \over x^2}\right) dx $$

But the geometric implications of this are that a cross-section of such an object has an infinite area but the volume is some finite value.

My questions:

1. Using an intuitive or geometric explanation, why doesn't $ 1\over x $ converge to some value?
2. Why is the volume described above finite while the cross-section is infinite?

Edit: Changed $[]$ to $()$

Answer 1

Suppose $\lim_{M\rightarrow\infty} \int_1^M \dfrac{dx}{x} $ exists. Letting $L$ be this limit, $\lim_{M\rightarrow\infty} \int_1^M \dfrac{dx}{x} =L$ so that, for any $c>0$ there is a $M(c)$ such that $0 \lt L-\int_1^M \dfrac{dx}{x} \lt c$ for $M > M(c)$. Choosing such an $M$, we also have $0 \lt L-\int_1^{2M} \dfrac{dx}{x} \lt c$ so that $0 \lt L-\int_1^{2M} \dfrac{dx}{x} =L-\int_1^{M} \dfrac{dx}{x}-\int_M^{2M} \dfrac{dx}{x} $ or $\int_M^{2M} \dfrac{dx}{x} \lt L-\int_1^{M} \dfrac{dx}{x} \lt c$.

But $\int_M^{2M} \dfrac{dx}{x} \gt \dfrac{M}{2M} =\dfrac12$.

This is a contradiction for $c < \dfrac12$.

This is, of course, a restating of the standard elementary proof that the harmonic sum diverges.

Answer 2

Here's something that may be helpful geometrically, borrowed from a classic argument about the harmonic series.

One argument for the divergence of the harmonic series goes as follows:

$$ \begin{align} &\frac12 + \frac13 + \frac14 + \frac15 + \frac16 + \frac17 + \frac18 + \dots \\ > &\frac12 + \frac14 + \frac14 + \frac18+ \frac18+ \frac18+ \frac18 + \dots \\ = &\frac12 + \left(\frac14 + \frac14\right) + \left(\frac18+ \frac18+ \frac18+ \frac18\right) + \dots \\ = &\frac12 + \frac12 + \frac12 + \dots \end{align} $$

Let's translate this into our integral. Imagine the area under the curve $\frac1x$. Fit a rectangle of area $\frac12$ between the points $(1,0),(2,0),(1,\frac12),(2,\frac12)$ -- this inside the area of the curve. Fit the next rectangle of area $\frac12$ between the points $(2,0),(4,0),(2,\frac14),(4,\frac14)$. In general the $i$th rectangle will be placed between the points $(2^{i-1},0),(2^i,0),(2^{i-1},\frac1{2^i}),(2^i,\frac1{2^i})$. No two rectangles overlap, and each rectangle has area $\frac12$. Since you can fit infinite rectangles of equal area under the integral, it must diverge.