# Does the series $\sum_{n=1}^{\infty}\frac{\sin(\cos(n))}{n}$ converge?

I was doing some exercises and this one just stunned me. I had to study the convergence of this serie:

$$\sum_{n=1}^{\infty}\frac{\sin(\cos(n))}{n}$$

I tried alot of diffrent things, but I got no where. Can anyone please help me with an idea or a clue just to start ?

• What exactly do you want to study in it? – The Dead Legend Apr 17 '17 at 23:29
• if it's converge or not – Hptunjy Prjkeizg Apr 17 '17 at 23:32
• Are you asking whether $\sum_{n=1}^\infty\sin(\cos(n))/n$ converges? – stewbasic Apr 17 '17 at 23:33
• This is an interesting question and honestly I do not understand the downvotes or the closing votes. – Jack D'Aurizio Apr 18 '17 at 3:11
• @SimplyBeautifulArt: of course, that is a matter of personal opinion. Mine is that an interesting question and a good formatting should grant for quality, so I am voting for reopening. Do we really want to be flooded by highly upvoted questions like prove that $\sum_{n\geq 1}\frac{1}{n^2 e^n e^{e^n}}$ is convergent and banish this kind of deeper math? – Jack D'Aurizio Apr 18 '17 at 14:01

Here is an alternate to Jack D'Aurizio's answer, which interestingly also involves the irrationality measure of $\pi$ (call it $\mu$). Suppose $f$ is a $2\pi$-periodic smooth function, such that $\int_0^{2\pi}f(x)\;dx=0$. Let $$s_n=\frac1n\sum_{k=1}^n f(k).$$ By the Koksma–Hlawka inequality and this bound, $$|s_n|=O(n^{-1/(\mu-1)+\epsilon})$$ for any $\epsilon>0$. In particular, choosing $\epsilon<\frac1{2(\mu-1)}$, we have $|s_n|=O(n^{-\epsilon})$. Using summation by parts, $$\begin{eqnarray*} \sum_{n=1}^N\frac{f(n)}n &=&s_N+\sum_{n=1}^{N-1}ns_n\left(\frac1n-\frac1{n+1}\right)\\ &=&s_N+\sum_{n=1}^{N-1}\frac{s_n}{n+1}. \end{eqnarray*}$$ The above implies $s_N\to0$ and the last sum is absolutely convergent, so the LHS converges. In particular setting $f(x)=\sin(\cos(x))$, the series in question converges.
I think it is more practical to expand $\sin\cos(x)$ as a Fourier cosine series. We have
$$\sin\cos(x) = 2\sum_{m\geq 0}(-1)^m J_{2m+1}(1) \cos((2m+1)x)$$ where the coefficients $J_{2m+1}(1)$, depending on a modified Bessel function of the first kind, have an exponential decay. On the other hand $$\sum_{n\geq 1}\frac{\cos(nx)}{n}=-\log\left|2\sin\frac{x}{2}\right|$$ for any $x\not\in 2\pi\mathbb{Z}$, and $-\log\left|2\sin\frac{x}{2}\right|$ cannot be too large for some small $x\in\mathbb{N}$ since $\pi$ has a finite irrationality measure. By exploiting the exponential decay of the previous coefficients we may conclude that the original series is convergent.
• you will loose the reader with a Bessel function, simply say $a_m$ decreases fast enough because $\sin \cos x$ is smooth ? – reuns Apr 18 '17 at 3:19
• @user1952009: of course, that and the irrationality measure of $\pi$ are the key ingredients. – Jack D'Aurizio Apr 18 '17 at 4:43