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Numerical experimentation suggests that the following limit converges.

$$L=\lim_{n\to\infty}{\dfrac{1}{n}\sum_{k=1}^n\sin{\left(\dfrac{n}{k}\right)}}$$

According to both Desmos and Wolfram, for $n=10^5, 10^6, 10^7$, the values of $L$ are $0.504116, 0.504069, 0.504068$, respectively.

Does this limit converge, and if so, is there a closed form?

By looking at the graph of $y=\sin{\left(\frac{n}{x}\right)}$ from $x=1$ to $x=n$, I can see why the limit should converge, but I don't know how to prove this rigorously. I doubt there is a closed form.

(In case you're wondering where this question comes from, I just made it up after thinking about another question.)

Kroki
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Dan
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2 Answers2

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For $g$ a bounded continuous function on $(0,1]$ (so integrable) let $g_n(x) = \sum_{k=1}^{n}\mathbf 1_{\left[\frac {k-1}n, \frac{k}{n} \right]}(x) g\left(\frac{k}{n}\right)$,

\begin{align} \left|g_n(x) - g(x)\right| &= \left|g\left(\frac{\left\lfloor nx\right\rfloor}{n}\right) - g(x)\right| \underset{n\to \infty}\to 0. \end{align}

Since $g_n$ are bounded by the $\sup\limits_{x\in (0,1]} \left|g(x)\right|$ then,

$$\int_0^1 g_n(x)\mathrm d x \underset{n\to \infty}\to \int_0^1 g(x)\mathrm d x$$

But $\displaystyle\int_{0}^1 g_n(x)\mathrm d x = \frac 1n\sum_{k=1}^n g\left(\frac kn\right)$.

Applying the previous result on $g(x) =\sin \left(\frac 1x\right)$, we have the sum converges to the integral $\displaystyle\int_0^1 \sin \left(\frac1x\right)\mathrm dx$. \begin{align} \int_{0}^1 \sin\left(\frac1x\right)\mathrm d x =_{t:=\frac1x} \int_{1}^\infty \frac{\sin(t)}{t^2}\mathrm d t = \left[-\frac{1}{t}\sin (t)\right]_0^{\infty}-\int_1^\infty \frac{\cos t}{t}\mathrm d = 1 - \int_{1}^{\infty} \frac{\cos t}{t}\mathrm d t. \end{align}

Kroki
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    I can't count how many times I've not realized that a series is a Riemann sum. Bars, darnit. – Dan Jul 29 '22 at 16:16
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    How do you know the Riemann sum converges to the integral for a case like this where it's an improper integral? – Daniel Schepler Jul 29 '22 at 16:25
  • @DanielSchepler good question! I will update my answer with an answer of that. – Kroki Jul 29 '22 at 16:31
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    It should not be called a Riemann sum. In the 19th century Riemann extended the classical integral to a wider class of functions with what is correctly called Riemann integration. – DanielWainfleet Jul 29 '22 at 16:33
  • Sorry about using the Riemann sum I will update my answer without using it. – Kroki Jul 29 '22 at 16:35
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    I would say the Riemann sum argument is at least a good heuristic, so you could keep it as an initial motivator, before you start going into more sensitive analysis. – Daniel Schepler Jul 29 '22 at 16:40
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    OK, now I see -- the $n$th value of the sequence in question is the integral of a simple function $f_n$, where $f_n(x) \to \sin(1/x)$ pointwise on $(0, 1]$ as $n \to \infty$ and each $|f_n(x)| \le 1$. Therefore, by the Dominated Convergence theorem, the sequence converges to the Lebesgue integral of $\int_0^1 \sin(1/x)dx$. – Daniel Schepler Jul 29 '22 at 17:51
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    Thanks for the edit! An upvote would be highly appreciated! –  Jul 29 '22 at 18:25
  • @DanielSchepler: the expression under limit in question is a Riemann sum for the function $f(x)=\sin(1/x)$. The function is Riemann integrable on $[0,1]$ so the limit exists. I don't think a deeper analysis is needed here. – Paramanand Singh Jul 30 '22 at 01:54
  • Also this is not a case of improper Riemann integral. Improper integral always involves either unbounded function or unbounded interval or both. The problem here can be treated simply by letting $f(x) =\sin(1/x),x\neq 0$ and $f(0)$ can be defined arbitrarily. – Paramanand Singh Jul 30 '22 at 01:56
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I think I got the solution which agrees with your approximations. We use the integral:

$\int_{1}^{x}\sin\dfrac{x}{t}dt$=$-x\displaystyle\int\limits_{x}^{1}\dfrac{\sin u}{u^{2}}du$=$x\displaystyle\int_{1}^{x}\dfrac{\sin u}{u^{2}}du$=

$x(Ci(x)-Ci(1)-\dfrac{\sin x}{x}+\sin(1))$.

Multiplied by $\dfrac{1}{x}$ we get $\displaystyle \lim_{x \to+\infty}$$(Ci(x)-Ci(1)-\dfrac{\sin x}{x}+ \sin(1))$.

It is known that $Ci(x)\to 0$ as $x\to+\infty$ and also $\dfrac{\sin x}{x}\to 0$.

So we are left with $-Ci(1)+\sin(1)$ which is approximately $ 0.50406732$ as you have calculated!

Kroki
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