Evaluate $\lim\limits_{n\to\infty}{\dfrac{1}{n}\sum_{k=1}^n\sin{\left(\dfrac{n}{k}\right)}}$.

Question

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.)

Answer 1

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

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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}

Answer 2

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!