In real analysis, there is a theorem known as the Lebesgue Dominated Convergence Theorem (LDCT), which is generally considered a very useful tool for determining whether the limit and the integral can be interchanged. The Lebesgue 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), where F(x) is integrable on E, then the limit and the integral 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 this theorem, but rather to talk about 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 some of my classmates. Although there were few responses, a few friends gave it a try. The result was: Students who have studied real analysis basically could not solve this problem, while those who had not studied real analysis and only knew basic calculus basically all solved it.
The method for this problem is very simple: calculate the integral directly to get \frac{1}{2}\arctan(n^2), and 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 the integral is bound to fail, because this integral and the limit are not interchangeable! Therefore, should we reflect on why friends who have learned more real analysis (especially those who just started) are instead unable to solve it? It is because they only think about the Dominated Convergence Theorem (or other related theorems) and fail to try the most primitive method—calculating it directly. Learning more things but being too constrained by form and textbooks—isn’t that a case of "learning more to no avail"?
Derived from this Dominated Convergence Theorem, there is another topic: is it actually difficult 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, for the problems we encounter in textbooks or exams, if we find it difficult to find a dominating function, I think it is most likely due to a lack of proficiency. When we took real analysis, the teacher said that dominating functions are hard to find, but the examples he gave were too weak to demonstrate this point. If students heard this from the teacher and then spread the word, I think that would not be ideal. 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 situation 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 use the Arithmetic-Geometric Mean (AM-GM) inequality to construct the dominating 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 so that the AM-GM inequality can be used to create the form we want. As for how to split it and why it should be split this way, the pattern is actually easy to discover, and readers should discover it for themselves. Among the "difficult" exercises we encounter, most belong to this type and can be handled with a unified method. However, if we look at 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) The technique using the AM-GM inequality will not succeed. In other words, it seems that for this type of problem, the inability to solve it using mean value inequalities is equivalent to the inability to find a dominating function (referring to the type where the denominator is a polynomial). This, of course, cannot be universally true, but there seems to be a very subtle relationship that readers need to perceive for themselves.
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, being constrained by form and by teachers really validates that phrase which shouldn’t be true: learning more is useless.
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