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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.