<s>[INST] Series $\sum \frac{\sin(n)}{n} \cdot \left(1+\cdots +\frac{1}{n}\right)$ convergence question I want to test the convergence of the series.  $$ \sum_{n=1}^{\infty} \frac{\sin(n)}{n} \cdot \left(1+\frac{1}{2} + \cdots + \frac{1}{n}\right)$$ My guess is this should diverge, and the below I provide the details
Note that 


*

*$\displaystyle 1+\frac{1}{2} + \cdots + \frac{1}{n} \geq 1 +\frac{1}{2} + \frac{1}{4} + \cdots + \frac{1}{2^{n-1}} = 2 \cdot \left(1-\frac{1}{2^{n}}\right)$

*So from above I conclude that $$\sum_{n=1}^{\infty} \frac{\sin(n)}{n} \cdot \left(1+\frac{1}{2}+\cdots + \frac{1}{n}\right) \geq  2 \cdot \sum_{n=1}^{\infty} \frac{\sin(n)}{n} - \cdot \underbrace{\sum_{n=1}^{\infty} \frac{\sin(n)}{n \cdot 2^{n}}}_{A}$$

*Now $A$ converges and I think $\displaystyle \sum \frac{\sin(n)}{n}$ diverges hence my given series should diverge. Am I correct in my reasoning?
 [/INST] The series is convergent and, for fun, I'll provide a derivation.
Let's first rewrite your sum as ($H_n$ is the $n$-th harmonic number) :
$$\tag{1}S:=\sum_{n=1}^{\infty} H_n\frac{\sin(n)}{n}=\Im\left(\sum_{n=1}^{\infty} H_n\frac{e^{in}}{n}\right)$$
Since $H_n\sim \log(n)\,$ as $\,n\to\infty$ (because the 'harmonic integral' corresponding to the arithmetic sum is the logarithm)
To evaluate $S$ let's use generating functions and suppose that the derivative of $f$ is defined by :
$$\tag{2}f'(x):=\sum_{n=1}^{\infty} H_n\;x^n=-\frac 1{1-x}\ln(1-x)$$
(expand $\frac 1{1-x}=1+x+x^2+\cdots\ $ and $\ -\ln(1-x)=x+\frac {x^2}2+\frac{x^3}3+\cdots$ and multiply !)
Integrating $(2)$ we get :
$$\tag{3}f(x)=\sum_{n=1}^{\infty} H_n \frac{x^{n+1}}{n+1}=-\int\frac{\ln(1-x)}{1-x} dx=\frac 12(\ln(1-x))^2$$
so that (using $\;H_n=H_{n+1}-\frac 1{n+1}\;$ and setting $\,k:=n+1\;$) :
$$\sum_{k=2}^{\infty} H_k \frac{x^k}k-\sum_{k=2}^{\infty} \frac{x^k}{k^2}=\sum_{k=1}^{\infty} H_k \frac{x^k}k-\sum_{k=1}^{\infty} \frac{x^k}{k^2}=\frac 12(\ln(1-x))^2$$
$\operatorname{Li}_2(x):=\sum_{k=1}^{\infty} \frac{x^k}{k^2}$ is the dilogarithm  but we prefer to use directly :
$$\tag{4}\sum_{k=1}^{\infty} H_k \frac{x^k}k=\sum_{k=1}^{\infty} \frac{x^k}{k^2}+\frac 12(\ln(1-x))^2$$
For $x=e^i$ in $(1)$ we get following expression of $S$ :
\begin{align}
S&=\Im\left(\sum_{k=1}^{\infty} H_k \frac{e^{ik}}k\right)\\
S&=\sum_{k=1}^{\infty} \frac{\sin(k)}{k^2}+\frac 12\Im\left(\ln\bigl(1-e^i\bigr)\right)^2\\
 S&=\boxed{\displaystyle\operatorname{Cl}_2(1)-\ln\left(2\sin\left(\frac 12\right)\right)\frac{\pi-1}2}\\
S&\approx 1.05895346485231034922735
\end{align}
with $\operatorname{Cl}_2$ the Clausen function  : $\displaystyle\operatorname{Cl}_2(x)=-\int_0^x \ln\left(2\sin\left(\frac t2\right)\right)dt$
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