What is the sum of the series
$$\frac {4}{10}+\frac {4\cdot7}{10\cdot20}+ \frac {4\cdot7\cdot10}{10\cdot20\cdot30}+\cdots?$$
I know how to check if a series is convergent or not.Is there any technique to find out the sum of a series like this where each term increases by a pattern?
My solution is adapted from another similar solution.
\begin{align} \sum_{n=1}^{\infty}\frac{4\cdot7\cdot\cdots\cdot(3n+1)}{n!10^n} &=\sum_{n=1}^{\infty}\frac{\frac43\cdot\frac73\cdot\cdots\cdot\frac{3n+1}3}{n!\left(10/3\right)^n}\\ &=\sum_{n=1}^{\infty}\binom{\frac{3n+1}{3}}{n}\left(\frac{3}{10}\right)^n\\ &=\left[\sum_{n=1}^{\infty}\binom{\frac{3n+1}{3}}{n}x^n\right]_{x=3/10}\\ &=\left[\sum_{n=1}^{\infty}\binom{-\frac{4}{3}}{n}(-x)^{n}\right]_{x=3/10}\\ &=\left[ \left(\left(1-x\right)^{-4/3}-1\right) \right]_{x=3/10}\\ &=\sqrt[3]{\left(\frac{10}{7}\right)^4}-1 \end{align}