I have tried evaluating this series 

$$\sum_{n=1}^{\infty}\frac{H_{n}^3}{(n+1)3^n} $$ 

using some methods but it's seems to me that it is very hard. However, I noticed that the series converges faster than the Riemann series.

**My question here is:**

Is there some mathematical technique for evaluating the above series?

**Note:** Here, $H_n$ denotes the harmonic numbers.
 
Thank you for any help.