The Lebesgue Dominated Convergence Theorem
In real analysis there is a well-known result, the Lebesgue dominated convergence theorem, which is generally regarded as a useful tool for deciding whether integration and taking limits can be interchanged. The dominated convergence theorem states that if a sequence of functions $\left\{f_n(x)\right\}$ defined on a set $E$ satisfies $|f_n(x)|\leq F(x)$, and $F(x)$ is integrable on $E$, then the integral and 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 post is not going to discuss the proof of the theorem, but rather some topics related to its application. First, let me invite interested readers to try the following problem:
$$\lim_{n\to\infty}\left(\int_0^1 \frac{n^2 x}{1+n^4 x^4}dx\right)$$more
A couple of days ago I posted this problem on QQ, and also asked some of my classmates about it. Not many people responded, but a few friends did give it a try. What I found was: students who had studied real analysis mostly couldn't solve it, while students who hadn't studied real analysis and only knew basic calculus mostly could.
The trick to this problem is very simple: just compute the integral directly, which gives $\frac{1}{2}\arctan(n^2)$, and then take the limit to get $\frac{\pi}{4}$. If a student who has studied real analysis racks their brains trying to find a dominating function, they are bound to hit a wall — any attempt to interchange the limit and the integral is doomed to fail, because in this case the integral and the limit cannot be interchanged! This should make us reflect: why is it that people who have studied more real analysis (especially those who just finished the course) end up unable to solve it? Because all they can think of is the dominated convergence theorem (or related theorems), and they never try the most elementary approach — actually computing the integral. Learning more, yet being overly bound by formalism and by textbooks — isn't that a case of "more learning, no benefit"?
There's another topic that follows from this dominated convergence theorem, namely: how hard is it, in practice, to find a dominating function? (Of course, this discussion presupposes that a dominating function actually exists.) For general functions, finding a dominating function is certainly quite hard. But for the kinds of problems we encounter in textbooks or exams, if we still find it hard to come up with a dominating function, I think the most likely explanation is that our technique simply isn't good enough. When I took real analysis, the instructor told us that dominating functions are hard to find, but the examples given were far too weak to actually support that claim. If a student hears the teacher say this and then just repeats it without further thought, I don't think that's a good thing. Below is an example that the instructor considered a case where finding the dominating function is relatively hard (requiring guesswork and then a case-by-case proof), whereas in fact this type of problem can be handled directly and uniformly by a single method.
$$\lim_{n\to\infty}\left(\int_0^1 \frac{n^3 x}{1+n^4 x^2}dx\right)$$
The textbook handles this by splitting into cases. In fact, all we need is the AM-GM inequality to construct the bounding function directly:
$$\begin{aligned}1+n^4 x^2&=1+\frac{1}{3}n^4 x^2+\frac{1}{3}n^4 x^2+\frac{1}{3}n^4 x^2\\ &\geq 4\sqrt[4]{1\times\frac{1}{3}n^4 x^2\times\frac{1}{3}n^4 x^2\times\frac{1}{3}n^4 x^2}\\ &=4\sqrt[4]{\frac{1}{27}}n^3 x^{3/2}\geq n^3 x^{3/2}\end{aligned}$$
Hence
$$\frac{n^3 x}{1+n^4 x^2}\leq \frac{n^3 x}{n^3 x^{3/2}} = x^{-1/2}$$
and $x^{-1/2}$ is integrable on $(0,1)$. This gives us the dominating function.
The method here is simply to split the denominator apart, and then use the AM-GM inequality to shape it into the form we want. As for how to split it, and why to split it that way, the pattern is actually easy to spot once you look for it — readers should discover it for themselves. Among the exercises we typically encounter, the ones that seem "hard" almost all belong to this type, and can all be handled by this same unified method. But for the problem at the beginning of this post,
$$\lim_{n\to\infty}\left(\int_0^1 \frac{n^2 x}{1+n^4 x^4}dx\right)$$
the AM-GM trick simply doesn't work. In other words, it seems that for this class of problems, being unable to solve them via AM-GM is equivalent to being unable to find a dominating function? (This refers specifically to the case where the denominator is a polynomial.) This certainly isn't a general rule, but there does seem to be some subtle connection here — I'll leave it to readers to sense it for themselves.
Readers might like to try applying the same technique to
$$\lim_{n\to\infty}\left(\int_0^1 \frac{(n x)^s}{1+(nx)^{s+1}}dx\right),\quad s > 0$$
In short, the point of this post is this: only when you truly digest and integrate what you've learned, forming your own way of understanding it, have you really learned something. Otherwise, if you remain bound by formalism and by what your teacher says, you really will end up proving true that saying which should never be true: "more learning, no benefit."
Translated automatically with claude-sonnet-5; all equations are reproduced verbatim from the source. Copyright remains with the original author.