Question:
This follow series have simple closed form? $$\sum_{n=1}^{\infty}\dfrac{1}{n^2H_{n}}$$ where $$H_{n}=1+\dfrac{1}{2}+\dfrac{1}{3}+\cdots+\dfrac{1}{n}$$
I suddenly thought of this series, because I know this series $$\dfrac{1}{n^2H_{n}}\approx\dfrac{1}{n^2\ln{n}}$$ since $$\lim_{n\to\infty}\dfrac{\dfrac{1}{n^2\ln{n}}}{\dfrac{1}{n^2}}=0$$ so $$\sum_{n=2}^{\infty}\dfrac{1}{n^2\ln{n}}$$ converges. So $$\sum_{n=1}^{\infty}\dfrac{1}{n^2H_{n}}$$ converges. But this sum have closed form? such as link sum
and Infinite Series $\sum\limits_{n=1}^\infty\frac{H_{2n+1}}{n^2}$ Thank you