Find the sum of the following infinite series Find the sum of the following infinite series in which numerator and denominator contains term which are product of integers in arithmetic progression:


$$\frac15+ \frac{1\times4}{5\times10}+\frac{1\times4\times7}{5\times10\times15}+\dots$$


I found this problem in an Indian competitive exam. I tried some conventional methods, but could not find the sum. Please offer any help you think is useful. Thanks!

Consider the series expansion for $f(x)=(1-x)^{-1/3}$:
$$\begin{align}f(x) &= 1+\frac13 x + \frac1{2!} \left ( -\frac13\right ) \left ( -\frac{4}{3}\right ) x^2 - \frac1{3!} \left ( -\frac13\right ) \left ( -\frac{4}{3}\right )\left ( -\frac{7}{3}\right ) x^3 +\cdots\\ &= 1+\frac13 x + \frac{1 \cdot 4}{2! 3^2} x^2 + \frac{1 \cdot 4 \cdot 7}{3! 3^3} x^2+\cdots\end{align}$$
Thus, the stated sum is simply
$$f \left ( \frac{3}{5} \right ) - 1 = \left ( \frac{5}{2} \right )^{1/3}-1$$
