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
$$f \left ( \frac{3}{5} \right ) - 1 = \left ( \frac{5}{2} \right )^{1/3}-1$$
Can be solved by looking here: $\frac {1}{5}-\frac{1.4}{5.10}+\frac{1.4.7}{5.10.15}-\cdot\cdot\cdot\cdot\cdot\cdot\cdot$
The sign change in the terms is covered by my answer.
