<s>[INST] $\lim_{n \to \infty}n^2\int_{1}^{\infty} \frac{cos(x/n)-1}{x^4}dx$ Show that the following limit exists and compute it: 
$$\lim_{n \to \infty}n^2\int_{1}^{\infty} \frac{\cos\left(\frac{x}{n}\right)-1}{x^4}\,dx$$

Attempt: By using the integration by parts, I get the following result: 
$$\begin{align*}\int_{1}^{\infty} \frac{\cos\left(\frac{x}{n}\right)-1}{x^4}\,dx&=\lim_{k\to\infty}\int_{1}^{k}\frac{\cos\left(\frac{x}{n}\right)-1}{x^4}\\\\
&=\lim_{k\to\infty}\left\{\left(\frac{k}{n}\right)^4\sin\left(\frac{k}{n}\right)+4\left(\frac{k}{n}\right)^3\cos\left(\frac{k}{n}\right)\right.\\
&\quad\quad\quad\quad-12\left(\frac{k}{n}\right)^2\sin\left(\frac{k}{n}\right)-24\left(\frac{k}{n}\right)\cos\left(\frac{k}{n}\right)\\
&\quad\quad\quad\quad+24\sin\left(\frac{k}{n}\right)+\frac{n^3}{3k^3}\\
&\quad\quad\quad\quad\left.-\left[\frac{1}{n^4}\sin\left(\frac{1}{n}\right)+\frac{4}{n^3}\cos\left(\frac{1}{n}\right)\right.\right.\\
&\quad\quad\quad\quad\left.\left.-12\left(\frac{1}{n}\right)^2\sin\left(\frac{1}{n}\right)-\frac{24}{n}\cos\left(\frac{1}{n}\right)\right.\right.\\
&\quad\quad\quad\quad\left.\left.+24\sin\left(\frac{1}{n}\right)+\frac{n^3}{3}\right]\right\}\end{align*}$$
But I could not compute this limit. And how can we show the existence of the limit $\displaystyle\lim_{n \to \infty}n^2\int_{1}^{\infty} \frac{\cos\left(\frac{x}{n}\right)-1}{x^4}\,dx$? Thanks!
 [/INST] $\newcommand{\abs}[1]{\left|#1\right|}
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\newcommand{\br}[1]{\left\{#1\right\}}$
I immediately thought about the dominated convergence theorem. If we move $n^2$ inside the integral, which we can, we get:
$$\int_1^\infty\dfrac{\cos(\frac{x}{n})-1}{x^4}n^2\mathrm{d}x.$$
For $n\to\infty$, the numerator is asymptotic to $-\frac{x^2}{2n^2}$, which means the whole is asymptotic to $-\frac{n^2x^2}{2n^2x^4}=-\frac{1}{2x^2}$, whose integral is $-\frac12$. Let's try to find a domination then. The first thing that came to my mind was:
$$\left|\frac{\cos(\frac{x}{n})-1}{x^4}\right|\leq\frac{|\cos(\frac{x}{n})|+1}{|x|^4}\leq\frac{2}{x^4}.$$
This is valid for any $n$, and this function is integrable over $[1,\infty)$ with respect to the Lebesgue measure. Of course, the question seems to be about a Riemann integral, and I'm talking about the Lebesgue integral. But then, since both exist (the domination grants the existence of the Lebesgue one, and if the Riemann one doesn't exist, well, the question is useless :)), they are equal, so if I prove things for the LI they are true for the RI as well. Well right, we have the domination, so we swap limit and integral, and we are left with the limit being:
$$-\int_1^\infty\frac{1}{2x^2}\mathrm{d}x=-\frac12.$$
Just out of curiosity, I will try evaluating the limit you couldn't compute:
\begin{align*}
\int_{1}^{\infty} \frac{\cos\pa{\frac{x}{n}}-1}{x^4}dx={}&\lim_{k\to\infty}\int_{1}^{k}\frac{\cos\pa{\frac{x}{n}}-1}{x^4}={} \\
{}={}&\lim_{k\to\infty}\left\{\pa{\frac{k}{n}}^4\sin\pa{\frac{k}{n}}+4\pa{\frac{k}{n}}^3\cos\pa{\frac{k}{n}}-{}\right. \\
&{}\left.{}+12\pa{\frac{k}{n}}^2\sin\pa{\frac{k}{n}}-24\pa{\frac{k}{n}}\cos\pa{\frac{k}{n}}+{}\right. \\
&{}\left.{}+24\sin\pa{\frac{k}{n}}+\pa{\frac{n^3}{3k^3}}-\left[\pa{\frac{1}{n^4}}\sin\pa{\frac1n}+\pa{\frac{4}{n^3}}\cos\pa{\frac1n}-{}\right.\right. \\
&{}\left.\left.{}+12\pa{\frac1n}^2\sin\pa{\frac1n}-\pa{\frac{24}{n}}\cos\pa{\frac1n}+24\sin\pa{\frac1n}+\frac{n^3}{3}\right]\right\}.
\end{align*}
First of all, there are a bunch of terms independent of $k$. I suggest we forget about those. Also, there is $\frac{n^3}{k^3}$ (or a constant times that), which obviously tends to 0 for $k\to\infty$. We take that away. Finally, the other terms all constantly have $\frac{k}{n}$, so I suggest we substitute it for $z=\frac{k}{n}$ to make the expression more "pretty". For $k\to\infty,z\to\infty$. Side note: do use operators (\sin, \cos) for sine and cosine; don't use the plain word in math mode; do use fractions; don't use the slash; do therefore lengthen the delimiters via \left…\right. So the limit becomes:
$$\lim_{z\to\infty}\br{z^4\sin z+4z^3\cos z-12z^2\sin z-24z\cos z+24\sin z}.$$
Please check your calculations, and mine, because one of us must have made a mistake, since the dominant term here is clearly the one for $z^4$ which has no limit. Yet the limit must exist, otherwise we would be trying to compute the limit of something that doesn't exist.
Update:
Following the reading of Jack's answer, I noticed I found a domiation for the integrand without $n^2$, and then applied the theorem to the integrals with $n^2$. Let me try another way. I reuse the first line of his answer by observing that:
$$n^2\frac{\cos(\frac{x}{n})-1}{x^4}=-2n^2\frac{\sin^2(\frac{x}{2n})}{x^4}=-2\pa{\frac{n\sin(\frac{x}{2n})}{x^2}}^2.$$
FOr a moment, we forget about the 2 and the square, and take the modulus instead. Let's try dominating that:
$$\left|\frac{n\sin\frac{x}{2n}}{x^2}\right|\leq\frac{n\frac{x}{2n}}{x^2}=\frac{1}{2x}.$$
That is not an integrable domination. However, now we remember we are integrating the square, so we apply the domination there, where it becomes $\frac{1}{4x^2}$ which is indeed integrable. So by dominated convergence theorem, the limit passes under integral, yielding:
\begin{align*}
\lim_{n\to\infty}\int_1^\infty n^2\frac{\cos(\frac{x}{n})-1}{x^4}\mathrm{d}x={}&-\lim_{n\to\infty}\int_1^\infty2\frac{n^2\sin^2(\frac{x}{2n})}{x^4}\mathrm{d}x=-\int_1^\infty\lim_{n\to\infty}2\frac{n^2\sin^2(\frac{x}{2n})}{x^4}\mathrm{d}x={} \\
{}={}&-2\cdot\hspace{-4pt}\int_1^\infty\frac{1}{4x^2}\mathrm{d}x=-\frac12,
\end{align*}
getting to the same result as before. I'm afraid Jack forgot to halve the argument of the sine when doing his equalities. I will check and notify him if so.
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