An interesting limit: $\lim_\limits{n\to\infty}\frac{\sin 1\sin\sqrt{1}+\sin 2\sin\sqrt{2}+\sin 3\sin\sqrt{3}+\cdots+\sin n\sin\sqrt{n}}{n}$

Question

I would like to prove that

$$\lim_\limits{n\to\infty}\frac{\sin 1\sin\sqrt{1}+\sin 2\sin\sqrt{2}+\sin 3\sin\sqrt{3}+\cdots+\sin n\sin\sqrt{n}}{n}=0$$

but I am stuck.

I tried to solve it by using Euler-Maclaurin formula, but I could not to.

Euler-Maclaurin formula applied to the function $f(x)=\sin x \sin\sqrt{x}\;\;$ is the following:

$$\sum_{h=1}^n\sin h\sin\sqrt{h}=\int_\limits{0}^n\left[\sin x\sin\sqrt{x}+\left(x-\lfloor x\rfloor\right)\left(\cos x\sin\sqrt{x}+\frac{\sin x\cos\sqrt{x}}{2\sqrt{x}}\right)\right] \, dx$$

but I could not manage to prove that

$$\frac{1}{n}\int_\limits{0}^n\left(x-\lfloor x\rfloor\right)\left(\cos x \sin\sqrt{x} \right) \, dx\rightarrow 0 \text{ as } n\to\infty.$$

Moreover I tried to write the limit as a limit of a Riemann sum, but I did not manage to.

Furthermore I tried to prove the following inequality:

$$\left|\sin 1\sin\sqrt{1}+\sin 2\sin\sqrt{2}+\cdots+\sin n \sin\sqrt{n} \right|\le\sqrt[4]{n^3}\\\text{for all }\;n\in\mathbb{N},$$

but it was not successful.

I managed to prove that

$$\lim_{n\to\infty}\frac{\sin 1+\sin 2 +\sin 3+\ldots+\sin n}{n}=0$$

and

$$\lim_{n\to\infty}\frac{\sin\sqrt{1}+\sin\sqrt{2}+\sin\sqrt{3}+\cdots+\sin\sqrt{n}}{n}=0.$$

Is it possible to use these last two limits in order to prove that

$$\lim_{n\to\infty}\frac{\sin 1\sin\sqrt{1}+\sin 2\sin\sqrt{2}+\sin 3 \sin\sqrt{3}+\cdots+\sin n\sin\sqrt{n}}{n}=0\text{ ?}$$

I tried to use Cauchy-Schwartz inequality, but I got $$\lim_{n\to\infty}\frac{\sin^21+\sin^22+\cdots+\sin^2n}{n}$$ and $$\lim_{n\to\infty}\frac{\sin^2\sqrt{1}+\sin^2\sqrt{2}+\cdots+\sin^2\sqrt{n}}{n}$$ and these last two limits are not zero in fact there are both $\frac{1}{2}$.

Answer 1

Let $S_n$ be given by

$$S_n=\sum_{k=1}^n \sin(k)\sin\sqrt{k}\tag1$$

Applying summation by parts to the sum in $(1)$ reveals

$$S_n=\sin(\sqrt {n+1})\sum_{k=1}^{n}\sin(k)-\sum_{k=1}^n \left(\sum_{\ell=1}^k \sin(\ell)\right)\left(\sin(\sqrt {k+1})-\sin(\sqrt{k})\right)\tag 2$$

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ESTIMATES:

The sum $\sum_{\ell=1}^k \sin(\ell)$ can be evaluated in closed form which provides the estimate

$$\begin{align} \left|\sum_{\ell=1}^n \sin(\ell)\right|&=\left|\csc(1/2)\sin(n/2)\sin((n+1)/2)\right|\\\\ \le \csc(1/2)\tag3 \end{align}$$

Moreover, from the Prosthaphaeresis identities, we have the estimate

$$\begin{align} \left|\sin(\sqrt {k+1})-\sin(\sqrt{k}\right|&=\left|\frac12\cos\left(\frac{\sqrt{k+1}+\sqrt{k}}{2}\right)\sin\left(\frac{\sqrt{k+1}-\sqrt{k}}{2}\right)\right|\\\\ &=\left|2\cos\left(\frac{\sqrt{k+1}+\sqrt{k}}{2}\right)\sin\left(\frac{1}{2(\sqrt{k+1}+\sqrt{k})}\right)\right|\\\\ &\le \frac{1}{\sqrt{k}}\tag4 \end{align}$$

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Using the estimates in $(3)$ and $(4)$ in $(2)$, we find that

$$\begin{align} |S_n|&\le \csc(1/2)\left(1+\sum_{k=1}^n\frac1{\sqrt k}\right)\\\\ &\le \csc(1/2)(1+2\sqrt n)\tag5 \end{align}$$

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Finally, using the estimate in $(5)$ we have

$$\left|\frac{S_n}{n}\right|\le \frac{\csc(1/2)(1+2\sqrt n)}{n}$$

whence application of the squeeze theorem recovers the coveted limit

$$\bbox[5px,border:2px solid #C0A000]{\lim_{n\to \infty}\frac{\sum_{k=1}^n \sin(k)\sin(\sqrt k)}{n}=0}$$

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NOTE: We have tacitly found that $$\limsup_{n\to \infty}\frac{S_n}{\sqrt n}\le 2\csc(1/2)$$

Answer 2

One can actually say more and show that:

$|\sin 1\sin\sqrt{1}+\sin 2\sin\sqrt{2}+\ldots+\sin n\sin\sqrt{n}| \le C$ for some universal constant.

Using the sine product formula, it is enough to prove the result for

$$C_1(n)=\cos(1+\sqrt{1})+\cos(2+\sqrt{2})+\cdots+\cos(n+\sqrt{n})$$ and

$$C_2(n)=\cos(1-\sqrt{1})+\cos(2-\sqrt{2})+\cdots+\cos(n-\sqrt{n})$$

and then taking real parts it is enough to show the result for

$$S_{1,2}(n)=\sum_{k=1}^ne^{i(k\pm\sqrt k)}$$

We will show that $|S_{1,2}| \le C$ for a universal constant $C$ so the result will follow and we will do the proof for $S_2$ indicating the estimate changes needed for $S_1$ which are minor.

Let $g(x)=\frac{x-\sqrt x}{2 \pi}, x \ge 1$ and note that $1/(4\pi) \le g'(x) \le 1/(2\pi)$ and this inequality is enough to prove our result (the fact that the lower and upper bounds are constants strictly between $0$ and $1$.

Note also that by omitting a fixed finite number of terms which we can bound trivially the result holds for functions $f(x)$ like $3x+100\sqrt x, -2x+x^{1-1/10000}$ and so on, the crucial part being that $g'(x)=f'(x)/(2\pi)=c+o(1), x \to \infty, c \ne 0, |c| <1$, so $0<c_1<|g'(x)|<c_2<1, x >k$ for constants $c_1,c_2,k$ and for the function $h$ involved in $S_1$ we have $1/(2\pi) \le |h'(x)| \le 3/(4\pi)$

Let $q(n)=g(n+1)-g(n), n \ge 1$ so by the MVT there is $n \le x_n \le n+1, q(n)=g'(x_n)$ In particular $q_n$ increasing since $g'$ does (if $g'$ would be decreasing like for $S_1$ we conjugate and replace $g$ by $-g$) and $1/(4\pi) \le q(n) \le 1/(2\pi)$

But now the identity:

$$e^{2\pi i g(k)}=1/2(1+i\cot \pi q(k))(e^{2\pi i g(k)}-e^{2\pi i g(k+1)})$$ gives that

$$S_2(n)=\sum_{k=1}^{n}e^{2\pi i g(k)}=\sum_1^{n}1/2(1+i\cot \pi q(k))(e^{2\pi i g(k)}-e^{2\pi i g(k+1)})=$$

$$=i/2\sum_{k=2}^{n-1}e^{2\pi i g(k)}(\cot \pi q(k)-\cot \pi q(k+1))+1/2(1+i\cot \pi q(1))e^{2\pi i g(1)}-(1/2)(1+i\cot \pi q(n))e^{2\pi i g(n+1)}$$

by rearranging the terms and noting that only terms with $g(1), g(n+1)$ appear only once

But now taking absolute values and noting that $\cot \pi q(k)-\cot \pi q(k+1)$ is decreasing since $1/4<\pi q(k) <1/2<\pi, q(k)$ increasing, we get:

$$|S_2(n)| \le 1/2 (\cot \pi q(2)-\cot \pi q(n))+1/2(|\cot \pi q(n)|+|\cot \pi q(1)|+1 \le C_2 $$ where $C_2$ is obtained by using that all the cotangtents above are at most $\cot 1/4$, so one can take $C_2=2\cot 1/4 +1$ for example and clearly we get a similar $C_1$ for $S_1$ so we are done!

Answer 3

Property 1:

If $\;\left\{a_n\right\}_{n\in\mathbb{N}}\;$ is a sequence of real numbers such that $\;\left\{a_n-a_{n-1}\right\}_{n\in\mathbb{N}\setminus\{1\}}\;$ is monotonic and there exists $\;k\in\mathbb{Z}\;$ for which $\;2\pi k<a_n-a_{n-1}<2\pi+2\pi k\;\;\;\;\forall n\in\mathbb{N}\setminus\{1\}\;,\;\;\;$ then $$\left|\sum_\limits{h=1}^n \cos a_h\right|\le\frac{1}{2}\left[\;\left|\cot\left(\frac{a_n-a_{n-1}}{2}\right)-\cot\left(\frac{a_2-a_1}{2}\right)\right|+\\+\left|\cot\left(\frac{a_n-a_{n-1}}{2}\right)\right|+|\sin a_1|\left|\cot\left(\frac{a_2-a_1}{2}\right)\right|+\\+|\cos a_1|+1\;\right]$$ for all $\;n\in\mathbb{N}\setminus\{1\}.$

Proof:

By applying Prosthaphaeresis identities, we get that

$\cos a_h+\cos a_{h+1}=2\cos\left(\frac{a_{h+1}+a_h}{2}\right)\cos\left(\frac{a_{h+1}-a_h}{2}\right)=\\=2\cos\left(\frac{a_{h+1}+a_h}{2}\right)\sin\left(\frac{a_{h+1}-a_h}{2}\right)\cot\left(\frac{a_{h+1}-a_h}{2}\right)=\\=\left(\sin a_{h+1}-\sin a_h\right)\cot\left(\frac{a_{h+1}-a_h}{2}\right)\;,\;\;\text{ for all }h\in\mathbb{N}.$

Moreover,

$2\sum_\limits{h=1}^n\cos a_h=\sum_\limits{h=1}^{n-1}\left(\cos a_h +\cos a_{h+1}\right)+\cos a_1+\cos a_n=\\=\sum_\limits{h=1}^{n-1}\left(\sin a_{h+1}-\sin a_h\right)\cot\left(\frac{a_{h+1}-a_h}{2}\right)+\cos a_1+\cos a_n =\\=\sum_\limits{h=1}^{n-1}\sin a_{h+1}\cot\left(\frac{a_{h+1}-a_h}{2}\right)-\sum_\limits{h=1}^{n-1}\sin a_h\cot\left(\frac{a_{h+1}-a_h}{2}\right)+\\+\cos a_1+\cos a_n=\\=\sum_\limits{h=2}^{n}\sin a_h\cot\left(\frac{a_h-a_{h-1}}{2}\right)-\sum_\limits{h=1}^{n-1}\sin a_h\cot\left(\frac{a_{h+1}-a_h}{2}\right)+\\+\cos a_1+\cos a_n =\\=\sum_\limits{h=2}^{n-1}\sin a_h\left[\cot\left(\frac{a_h-a_{h-1}}{2}\right)-\cot\left(\frac{a_{h+1}-a_h}{2}\right)\right]+\\+\sin a_n\cot\left(\frac{a_n-a_{n-1}}{2}\right)-\sin a_1\cot\left(\frac{a_2-a_1}{2}\right)+\cos a_1+\cos a_n\;,\\\text{ for all }\;n\in\mathbb{N}\setminus\{1\}.$

Since the function $\;\cot\;$ is monotonic on $\;\left]\pi k,\pi+\pi k\right[\;$ and $\;\left\{a_n-a_{n-1}\right\}_{n\in\mathbb{N}\setminus\{1\}}\;$ is a monotonic sequence such that $\;2\pi k<a_n-a_{n-1}<2\pi+2\pi k\;\;\;\;\forall n\in\mathbb{N}\setminus\{1\}\;,\;$ then the sequence $\;\left\{\cot\left(\frac{a_n-a_{n-1}}{2}\right)\right\}_{n\in\mathbb{N}\setminus\{1\}}\;$ is monotonic too.

So by taking absolute values and by noting that the sequence $\;\left\{\cot\left(\frac{a_n-a_{n-1}}{2}\right)\right\}_{n\in\mathbb{N}\setminus\{1\}}\;$ is monotonic, we get that

$2\left|\sum_\limits{h=1}^n\cos a_h\right|\le\sum_\limits{h=2}^{n-1}\left|\cot\left(\frac{a_h-a_{h-1}}{2}\right)-\cot\left(\frac{a_{h+1}-a_h}{2}\right)\right|+\\+\left|\cot\left(\frac{a_n-a_{n-1}}{2}\right)\right|+|\sin a_1|\left|\cot\left(\frac{a_2-a_1}{2}\right)\right|+|\cos a_1|+1=\\=\left|\cot\left(\frac{a_2-a_1}{2}\right)-\cot\left(\frac{a_n-a_{n-1}}{2}\right)\right|+\left|\cot\left(\frac{a_n-a_{n-1}}{2}\right)\right|+\\+|\sin a_1|\left|\cot\left(\frac{a_2-a_1}{2}\right)\right|+|\cos a_1|+1=\\=\left|\cot\left(\frac{a_n-a_{n-1}}{2}\right)-\cot\left(\frac{a_2-a_1}{2}\right)\right|+\left|\cot\left(\frac{a_n-a_{n-1}}{2}\right)\right|+\\+|\sin a_1|\left|\cot\left(\frac{a_2-a_1}{2}\right)\right|+|\cos a_1|+1\;,$

for all $\;n\in\mathbb{N}\setminus\{1\}.$

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Property 2:

If $\;\left\{a_n\right\}_{n\in\mathbb{N}}\;$ is a sequence of real numbers such that $\;\left\{a_n-a_{n-1}\right\}_{n\in\mathbb{N}\setminus\{1\}}\;$ is monotonic and there exists $\;k\in\mathbb{Z}\;$ for which $\;2\pi k<a_n-a_{n-1}<2\pi+2\pi k\;\;\;\;\forall n\in\mathbb{N}\setminus\{1\}\;,\;\;\;$ then $$\left|\sum_\limits{h=1}^n \sin a_h\right|\le\frac{1}{2}\left[\; \left|\cot\left(\frac{a_n-a_{n-1}}{2}\right)-\cot\left(\frac{a_2-a_1}{2}\right)\right|+\\+\left|\cot\left(\frac{a_n-a_{n-1}}{2}\right)\right|+|\cos a_1|\left|\cot\left(\frac{a_2-a_1}{2}\right)\right|+\\+|\sin a_1|+1\;\right]$$ for all $\;n\in\mathbb{N}\setminus\{1\}.$

Proof:

By applying Prosthaphaeresis identities, we get that

$\sin a_h+\sin a_{h+1}=2\sin\left(\frac{a_{h+1}+a_h}{2}\right)\cos\left(\frac{a_{h+1}-a_h}{2}\right)=\\=2\sin\left(\frac{a_{h+1}+a_h}{2}\right)\sin\left(\frac{a_{h+1}-a_h}{2}\right)\cot\left(\frac{a_{h+1}-a_h}{2}\right)=\\=\left(\cos a_h-\cos a_{h+1}\right)\cot\left(\frac{a_{h+1}-a_h}{2}\right)\;,\;\;\text{ for all }h\in\mathbb{N}.$

Moreover,

$2\sum_\limits{h=1}^n\sin a_h=\sum_\limits{h=1}^{n-1}\left(\sin a_h +\sin a_{h+1}\right)+\sin a_1+\sin a_n=\\=\sum_\limits{h=1}^{n-1}\left(\cos a_h-\cos a_{h+1}\right)\cot\left(\frac{a_{h+1}-a_h}{2}\right)+\sin a_1+\sin a_n =\\=\sum_\limits{h=1}^{n-1}\cos a_h\cot\left(\frac{a_{h+1}-a_h}{2}\right)-\sum_\limits{h=1}^{n-1}\cos a_{h+1}\cot\left(\frac{a_{h+1}-a_h}{2}\right)+\\+\sin a_1+\sin a_n=\\=\sum_\limits{h=1}^{n-1}\cos a_h\cot\left(\frac{a_{h+1}-a_h}{2}\right)-\sum_\limits{h=2}^n\cos a_h\cot\left(\frac{a_h-a_{h-1}}{2}\right)+\\+\sin a_1+\sin a_n =\\=\sum_\limits{h=2}^{n-1}\cos a_h\left[\cot\left(\frac{a_{h+1}-a_h}{2}\right)-\cot\left(\frac{a_h-a_{h-1}}{2}\right)\right]+\\+\cos a_1\cot\left(\frac{a_2-a_1}{2}\right)-\cos a_n\cot\left(\frac{a_n-a_{n-1}}{2}\right)+\sin a_1+\sin a_n\;,\\\text{ for all }\;n\in\mathbb{N}\setminus\{1\}.$

Since the function $\;\cot\;$ is monotonic on $\;\left]\pi k,\pi+\pi k\right[\;$ and $\;\left\{a_n-a_{n-1}\right\}_{n\in\mathbb{N}\setminus\{1\}}\;$ is a monotonic sequence such that $\;2\pi k<a_n-a_{n-1}<2\pi+2\pi k\;\;\;\;\forall n\in\mathbb{N}\setminus\{1\}\;,\;$ then the sequence $\;\left\{\cot\left(\frac{a_n-a_{n-1}}{2}\right)\right\}_{n\in\mathbb{N}\setminus\{1\}}\;$ is monotonic too.

So by taking absolute values and by noting that the sequence $\;\left\{\cot\left(\frac{a_n-a_{n-1}}{2}\right)\right\}_{n\in\mathbb{N}\setminus\{1\}}\;$ is monotonic, we get that

$2\left|\sum_\limits{h=1}^n\sin a_h\right|\le\sum_\limits{h=2}^{n-1}\left|\cot\left(\frac{a_{h+1}-a_h}{2}\right)-\cot\left(\frac{a_h-a_{h-1}}{2}\right)\right|+\\+|\cos a_1|\left|\cot\left(\frac{a_2-a_1}{2}\right)\right|+\left|\cot\left(\frac{a_n-a_{n-1}}{2}\right)\right|+|\sin a_1|+1=\\=\left|\cot\left(\frac{a_n-a_{n-1}}{2}\right)-\cot\left(\frac{a_2-a_1}{2}\right)\right|+\left|\cot\left(\frac{a_n-a_{n-1}}{2}\right)\right|+\\+|\cos a_1|\left|\cot\left(\frac{a_2-a_1}{2}\right)\right|+|\sin a_1|+1\;,$

for all $\;n\in\mathbb{N}\setminus\{1\}.$

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Corollary 1:

The sequences $\;\left\{\alpha_n=n+\sqrt{n}\right\}_{n\in\mathbb{N}}\;$ and $\left\{\beta_n=n-\sqrt{n}\right\}_{n\in\mathbb{N}}\;$ satisfy all the hypothesis of the previous properties and

$\left|\sum_\limits{h=1}^n \cos\left(h+\sqrt{h}\right)\right|<\frac{5}{2}\;,\;\;\text{ for all }\;n\in\mathbb{N}\;,$

$\left|\sum_\limits{h=1}^n \sin\left(h+\sqrt{h}\right)\right|<\frac{5}{2}\;,\;\;\text{ for all }\;n\in\mathbb{N}\;,$

$\left|\sum_\limits{h=1}^n \cos\left(h-\sqrt{h}\right)\right|<\frac{8}{3}\;,\;\;\text{ for all }\;n\in\mathbb{N}\;,$

$\left|\sum_\limits{h=1}^n \sin\left(h-\sqrt{h}\right)\right|<\frac{23}{6}\;,\;\;\text{ for all }\;n\in\mathbb{N}.$

Proof:

$\alpha_n-\alpha_{n-1}=n+\sqrt{n}-n+1-\sqrt{n-1}=\\=1+\sqrt{n}-\sqrt{n-1}=1+\frac{1}{\sqrt{n}+\sqrt{n-1}}\;,\\\text{for all }\;n\in\mathbb{N}\setminus\{1\}.$

Hence the sequence $\;\left\{\alpha_n-\alpha_{n-1}\right\}_{n\in\mathbb{N}\setminus\{1\}}\;$ is monotonically decreasing and $\;0<1<\alpha_n-\alpha_{n-1}\le\sqrt{2}<\pi<2\pi\;,$

$\text{for all }\;n\in\mathbb{N}\setminus\{1\}.$

Since the function $\;\cot\;$ is monotonically decreasing on $\;\left]0,\pi\right[\;$ and $\;\left\{\alpha_n-\alpha_{n-1}\right\}_{n\in\mathbb{N}\setminus\{1\}}\;$ is a decreasing sequence such that $\;0<\alpha_n-\alpha_{n-1}<2\pi\;\;\;\;\forall n\in\mathbb{N}\setminus\{1\}\;,\;$ then the sequence $\;\left\{\cot\left(\frac{\alpha_n-\alpha_{n-1}}{2}\right)\right\}_{n\in\mathbb{N}\setminus\{1\}}\;$ is monotonically increasing.

By applying the property $1$, we get that

$\left|\sum_\limits{h=1}^n \cos\alpha_h\right|\le\frac{1}{2}\left[\;\left|\cot\left(\frac{\alpha_n-\alpha_{n-1}}{2}\right)-\cot\left(\frac{\alpha_2-\alpha_1}{2}\right)\right|+\\+\left|\cot\left(\frac{\alpha_n-\alpha_{n-1}}{2}\right)\right|+|\sin \alpha_1|\left|\cot\left(\frac{\alpha_2-\alpha_1}{2}\right)\right|+|\cos\alpha_1|+1\;\right]=\\=\frac{1}{2}\left[\;\cot\left(\frac{\alpha_n-\alpha_{n-1}}{2}\right)-\cot\left(\frac{\sqrt{2}}{2}\right)+\\+\cot\left(\frac{\alpha_n-\alpha_{n-1}}{2}\right)+\sin 2\cot\left(\frac{\sqrt{2}}{2}\right)-\cos 2+1\;\right]=\\=\cot\left(\frac{\alpha_n-\alpha_{n-1}}{2}\right)+\frac{1}{2}\left(\sin 2-1\right)\cot\left(\frac{\sqrt{2}}{2}\right)+\frac{1}{2}\left(1-\cos 2\right)<\\<\cot\left(\frac{1}{2}\right)+\frac{1}{2}\left(\sin 2-1\right)\cot\left(\frac{\sqrt{2}}{2}\right)+\frac{1}{2}\left(1-\cos 2\right)<\frac{5}{2}\;,$

for all $\;n\in\mathbb{N}\setminus\{1\}.$

Therefore,

$\left|\sum_\limits{h=1}^n \cos\left(h+\sqrt{h}\right)\right|<\frac{5}{2}\;,\;\;\text{ for all }\;n\in\mathbb{N}.$

And by applying the property $2$, we get that

$\left|\sum_\limits{h=1}^n \sin\alpha_h\right|\le\frac{1}{2}\left[\;\left|\cot\left(\frac{\alpha_n-\alpha_{n-1}}{2}\right)-\cot\left(\frac{\alpha_2-\alpha_1}{2}\right)\right|+\\+\left|\cot\left(\frac{\alpha_n-\alpha_{n-1}}{2}\right)\right|+|\cos \alpha_1|\left|\cot\left(\frac{\alpha_2-\alpha_1}{2}\right)\right|+|\sin\alpha_1|+1\;\right]=\\=\frac{1}{2}\left[\;\cot\left(\frac{\alpha_n-\alpha_{n-1}}{2}\right)-\cot\left(\frac{\sqrt{2}}{2}\right)+\\+\cot\left(\frac{\alpha_n-\alpha_{n-1}}{2}\right)-\cos 2\cot\left(\frac{\sqrt{2}}{2}\right)+\sin 2+1\;\right]=\\=\cot\left(\frac{\alpha_n-\alpha_{n-1}}{2}\right)-\frac{1}{2}\left(1+\cos 2\right)\cot\left(\frac{\sqrt{2}}{2}\right)+\frac{1}{2}\left(1+\sin 2\right)<\\<\cot\left(\frac{1}{2}\right)-\frac{1}{2}\left(1+\cos 2\right)\cot\left(\frac{\sqrt{2}}{2}\right)+\frac{1}{2}\left(1+\sin 2\right)<\frac{5}{2}\;,$

for all $\;n\in\mathbb{N}\setminus\{1\}.$

Therefore,

$\left|\sum_\limits{h=1}^n \sin\left(h+\sqrt{h}\right)\right|<\frac{5}{2}\;,\;\;\text{ for all }\;n\in\mathbb{N}.$

Moreover,

$\beta_n-\beta_{n-1}=n-\sqrt{n}-n+1+\sqrt{n-1}=\\=1-\sqrt{n}+\sqrt{n-1}=1-\frac{1}{\sqrt{n}+\sqrt{n-1}}\;,\\\text{for all }\;n\in\mathbb{N}\setminus\{1\}.$

Hence the sequence $\;\left\{\beta_n-\beta_{n-1}\right\}_{n\in\mathbb{N}\setminus\{1\}}\;$ is monotonically increasing and $\;0<2-\sqrt{2}\le\beta_n-\beta_{n-1}<1<\pi<2\pi\;,$

$\text{for all }\;n\in\mathbb{N}\setminus\{1\}.$

Since the function $\;\cot\;$ is monotonically decreasing on $\;\left]0,\pi\right[\;$ and $\;\left\{\beta_n-\beta_{n-1}\right\}_{n\in\mathbb{N}\setminus\{1\}}\;$ is an increasing sequence such that $\;0<\beta_n-\beta_{n-1}<2\pi\;\;\;\;\forall n\in\mathbb{N}\setminus\{1\}\;,\;$ then the sequence $\;\left\{\cot\left(\frac{\beta_n-\beta_{n-1}}{2}\right)\right\}_{n\in\mathbb{N}\setminus\{1\}}\;$ is monotonically decreasing.

By applying the property $1$, we get that

$\left|\sum_\limits{h=1}^n \cos\beta_h\right|\le\frac{1}{2}\left[\;\left|\cot\left(\frac{\beta_n-\beta_{n-1}}{2}\right)-\cot\left(\frac{\beta_2-\beta_1}{2}\right)\right|+\\+\left|\cot\left(\frac{\beta_n-\beta_{n-1}}{2}\right)\right|+|\sin \beta_1|\left|\cot\left(\frac{\beta_2-\beta_1}{2}\right)\right|+|\cos\beta_1|+1\;\right]=\\=\frac{1}{2}\left[\;\cot\left(\frac{2-\sqrt{2}}{2}\right)-\cot\left(\frac{\beta_n-\beta_{n-1}}{2}\right)+\\+\cot\left(\frac{\beta_n-\beta_{n-1}}{2}\right)+\sin 0\cot\left(\frac{2-\sqrt{2}}{2}\right)+\cos0+1\;\right]=\\=1+\frac{1}{2}\cot\left(\frac{2-\sqrt{2}}{2}\right)<\frac{8}{3}\;,$

for all $\;n\in\mathbb{N}\setminus\{1\}.$

Therefore,

$\left|\sum_\limits{h=1}^n \cos\left(h-\sqrt{h}\right)\right|<\frac{8}{3}\;,\;\;\text{ for all }\;n\in\mathbb{N}.$

And by applying the property $2$, we get that

$\left|\sum_\limits{h=1}^n \sin\beta_h\right|\le\frac{1}{2}\left[\;\left|\cot\left(\frac{\beta_n-\beta_{n-1}}{2}\right)-\cot\left(\frac{\beta_2-\beta_1}{2}\right)\right|+\\+\left|\cot\left(\frac{\beta_n-\beta_{n-1}}{2}\right)\right|+|\cos\beta_1|\left|\cot\left(\frac{\beta_2-\beta_1}{2}\right)\right|+|\sin\beta_1|+1\;\right]=\\=\frac{1}{2}\left[\;\cot\left(\frac{2-\sqrt{2}}{2}\right)-\cot\left(\frac{\beta_n-\beta_{n-1}}{2}\right)+\\+\cot\left(\frac{\beta_n-\beta_{n-1}}{2}\right)+\cos 0\cot\left(\frac{2-\sqrt{2}}{2}\right)+\sin 0+1\;\right]=\\=\frac{1}{2}+\cot\left(\frac{2-\sqrt{2}}{2}\right)<\frac{23}{6}\;,$

for all $\;n\in\mathbb{N}\setminus\{1\}.$

Therefore,

$\left|\sum_\limits{h=1}^n \sin\left(h-\sqrt{h}\right)\right|<\frac{23}{6}\;,\;\;\text{ for all }\;n\in\mathbb{N}.$

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Corollary 2:

$\left|\sum_\limits{h=1}^n \sin h\sin\sqrt{h}\right|<\frac{31}{12}<\frac{13}{5}\;,\;\;\text{ for all }\;n\in\mathbb{N}\;,$

$\left|\sum_\limits{h=1}^n \cos h\cos\sqrt{h}\right|<\frac{31}{12}<\frac{13}{5}\;,\;\;\text{ for all }\;n\in\mathbb{N}\;,$

$\left|\sum_\limits{h=1}^n \sin h\cos\sqrt{h}\right|<\frac{19}{6}<\frac{16}{5}\;,\;\;\text{ for all }\;n\in\mathbb{N}\;,$

$\left|\sum_\limits{h=1}^n \cos h\sin\sqrt{h}\right|<\frac{19}{6}<\frac{16}{5}\;,\;\;\text{ for all }\;n\in\mathbb{N}.$

Proof:

Using the results of Corollary 1, we get that

$\left|\sum_\limits{h=1}^n \sin h\sin\sqrt{h}\right|=\frac{1}{2}\left|\sum_\limits{h=1}^n \left[\cos\left(h-\sqrt{h}\right)-\cos\left(h+\sqrt{h}\right)\right]\right|=\\=\frac{1}{2}\left|\sum_\limits{h=1}^n\cos\left(h-\sqrt{h}\right)-\sum_\limits{h=1}^n\cos\left(h+\sqrt{h}\right)\right|\le\\\le\frac{1}{2}\left[\;\left|\sum_\limits{h=1}^n\cos\left(h-\sqrt{h}\right)\right|+\left|\sum_\limits{h=1}^n\cos\left(h+\sqrt{h}\right)\right|\;\right]<\\<\frac{1}{2}\left[\frac{8}{3}+\frac{5}{2}\right]=\frac{31}{12}<\frac{13}{5}\;,\;\;\;\;\text{ for all }\;n\in\mathbb{N}\;,$

$\left|\sum_\limits{h=1}^n \cos h\cos\sqrt{h}\right|=\frac{1}{2}\left|\sum_\limits{h=1}^n \left[\cos\left(h+\sqrt{h}\right)+\cos\left(h-\sqrt{h}\right)\right]\right|=\\=\frac{1}{2}\left|\sum_\limits{h=1}^n\cos\left(h+\sqrt{h}\right)+\sum_\limits{h=1}^n\cos\left(h-\sqrt{h}\right)\right|\le\\\le\frac{1}{2}\left[\;\left|\sum_\limits{h=1}^n\cos\left(h+\sqrt{h}\right)\right|+\left|\sum_\limits{h=1}^n\cos\left(h-\sqrt{h}\right)\right|\;\right]<\\<\frac{1}{2}\left[\frac{5}{2}+\frac{8}{3}\right]=\frac{31}{12}<\frac{13}{5}\;,\;\;\;\;\text{ for all }\;n\in\mathbb{N}\;,$

$\left|\sum_\limits{h=1}^n \sin h\cos\sqrt{h}\right|=\frac{1}{2}\left|\sum_\limits{h=1}^n \left[\sin\left(h+\sqrt{h}\right)+\sin\left(h-\sqrt{h}\right)\right]\right|=\\=\frac{1}{2}\left|\sum_\limits{h=1}^n\sin\left(h+\sqrt{h}\right)+\sum_\limits{h=1}^n\sin\left(h-\sqrt{h}\right)\right|\le\\\le\frac{1}{2}\left[\;\left|\sum_\limits{h=1}^n\sin\left(h+\sqrt{h}\right)\right|+\left|\sum_\limits{h=1}^n\sin\left(h-\sqrt{h}\right)\right|\;\right]<\\<\frac{1}{2}\left[\frac{5}{2}+\frac{23}{6}\right]=\frac{19}{6}<\frac{16}{5}\;,\;\;\;\;\text{ for all }\;n\in\mathbb{N}\;,$

$\left|\sum_\limits{h=1}^n \cos h\sin\sqrt{h}\right|=\frac{1}{2}\left|\sum_\limits{h=1}^n \left[\sin\left(h+\sqrt{h}\right)-\sin\left(h-\sqrt{h}\right)\right]\right|=\\=\frac{1}{2}\left|\sum_\limits{h=1}^n\sin\left(h+\sqrt{h}\right)-\sum_\limits{h=1}^n\sin\left(h-\sqrt{h}\right)\right|\le\\\le\frac{1}{2}\left[\;\left|\sum_\limits{h=1}^n\sin\left(h+\sqrt{h}\right)\right|+\left|\sum_\limits{h=1}^n\sin\left(h-\sqrt{h}\right)\right|\;\right]<\\<\frac{1}{2}\left[\frac{5}{2}+\frac{23}{6}\right]=\frac{19}{6}<\frac{16}{5}\;,\;\;\;\;\text{ for all }\;n\in\mathbb{N}\;.$

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Corollary 3:

$\lim_\limits{n\to\infty}\frac{\sin 1\sin\sqrt{1}+\sin 2\sin\sqrt{2}+\sin 3\sin\sqrt{3}+\ldots+\sin n\sin\sqrt{n}}{n^\gamma}=0$

for any $\;\gamma>0.$

Proof:

Since $$-\frac{31}{12 n^\gamma}<\frac{\sum_\limits{h=1}^n \sin h\sin\sqrt{h}}{n^\gamma}<\frac{31}{12 n^\gamma}\;\;\;\;\text{ for all }\;n\in\mathbb{N}$$ and $\;\lim_\limits{n\to\infty}\left(-\frac{31}{12 n^\gamma}\right)=0\;,\;\;\lim_\limits{n\to\infty}\frac{31}{12 n^\gamma}=0\;,$

by applying the squeeze theorem, we get that

$\lim_\limits{n\to\infty}\frac{\sin 1\sin\sqrt{1}+\sin 2\sin\sqrt{2}+\sin 3\sin\sqrt{3}+\ldots+\sin n\sin\sqrt{n}}{n^\gamma}=0\;.$