By 苏剑林 | January 16, 2015
In Real Analysis, there is Lebesgue's Dominated Convergence Theorem, which is generally considered a very useful method for determining whether integration and taking limits can be interchanged. The Dominated Convergence Theorem states that if a sequence of functions $\{f_n(x)\}$ defined on a set $E$ satisfies $|f_n(x)| \leq F(x)$, and $F(x)$ is integrable on $E$, then the integration and taking the limit can be interchanged, i.e.,
$$ \lim_{n \to \infty} \left( \int_E f_n(x)dx \right) = \int_E \left( \lim_{n \to \infty} f_n(x) \right) dx $$
This article does not intend to discuss the proof of the theorem, but rather the topics related to its application. First, for interested readers, please try the following problem:
$$ \lim_{n \to \infty} \left( \int_0^1 \frac{n^2 x}{1+n^4 x^4} dx \right) $$
A few days ago, I posted this problem on QQ and asked the classmates around me. There were few responses, but several friends attempted it. The result was: students who had studied Real Analysis generally could not solve this problem, while students who had not studied Real Analysis but had only studied basic Calculus basically all solved it.
The method for this problem is very simple: directly evaluate the integral to get $\frac{1}{2}\arctan(n^2)$, then take the limit to get $\frac{\pi}{4}$. If a student who has studied Real Analysis desperately tries to find a dominating function, they will inevitably hit a wall. Any attempt to interchange the limit and integration is destined to fail because the integration and limit for this specific problem are not interchangeable! Therefore, should we reflect on why friends who have learned more Real Analysis (especially those who just learned it) are instead unable to solve it? It is because they only think about the Dominated Convergence Theorem (or other related theorems) and forget to try the most primitive method—calculating it. Having learned more things but being too rigid in form and tethered to textbooks—isn't this a case of "learning more to no benefit"?
Extended from this Dominated Convergence Theorem is another topic: how difficult is it to find a dominating function (of course, this question is discussed under the premise that a dominating function exists). First, for general functions, a dominating function is certainly quite difficult to find. However, regarding the problems we encounter in books or exams, if we find the dominating function difficult to find, I think it is most likely due to a lack of mastery. When we took Real Analysis, the teacher said that dominating functions are hard to find, but the examples he gave were too weak to actually demonstrate that point. If students took that sentence from the teacher and spread it around, I think that wouldn't be very good. Below is an example that the teacher considered difficult for finding a dominating function (requiring guessing and then piecewise proof), but in fact, this type of case can be handled using a direct and unified method to obtain the dominating function:
$$ \lim_{n \to \infty} \left( \int_0^1 \frac{n^3 x}{1+n^4 x^2} dx \right) $$
The textbook uses piecewise discussion. In fact, one only needs to utilize the Arithmetic-Geometric Mean (AM-GM) inequality to construct the function:
\begin{aligned}
1+n^4 x^2 &= 1 + \frac{1}{3}n^4 x^2 + \frac{1}{3}n^4 x^2 + \frac{1}{3}n^4 x^2 \\
&\geq 4\sqrt[4]{1 \times \frac{1}{3}n^4 x^2 \times \frac{1}{3}n^4 x^2 \times \frac{1}{3}n^4 x^2} \\
&= 4\sqrt[4]{\frac{1}{27}}n^3 x^{3/2} \geq n^3 x^{3/2}
\end{aligned}
Therefore:
$$ \frac{n^3 x}{1+n^4 x^2} \leq \frac{n^3 x}{n^3 x^{3/2}} = x^{-1/2} $$
Since $x^{-1/2}$ is integrable on $(0,1)$, we have found the dominating function.
This method involves splitting the denominator and then using the AM-GM inequality to achieve the form we want. As for how to split it and why it is split this way, it is actually quite easy to discover the pattern; readers should discover it for themselves. Among the practice problems we encounter, those considered "difficult" mostly belong to this type and can be handled with a unified method. However, if it were the problem at the beginning of the article:
$$ \lim_{n \to \infty} \left( \int_0^1 \frac{n^2 x}{1+n^4 x^4} dx \right) $$
Using the AM-GM inequality technique will not succeed. Does this mean that for such problems (referring to the type where the denominator is a polynomial), if it cannot be solved with the mean inequality, it is equivalent to not being able to find a dominating function? This, of course, isn't universally true, but there seems to be a very subtle relationship that requires the reader's own perception.
Readers might as well use the same technique to try:
$$ \lim_{n \to \infty} \left( \int_0^1 \frac{(n x)^s}{1+(nx)^{s+1}} dx \right), \quad s > 0 $$
In summary, this article aims to point out: only by integrating what you have learned and forming your own way of understanding can you be considered to have truly learned something. Otherwise, by sticking too rigidly to forms and teachers, you would truly fall into the sentiment that shouldn't hold true: learning more is useless.