The infinite sum
$$ 1+ \frac{1}{3} + \frac{1\cdot3}{3\cdot6} + \frac{1\cdot3\cdot5}{3\cdot6\cdot9} + \frac{1\cdot3\cdot5\cdot7}{3\cdot6\cdot9\cdot12} +\dots $$ is
a.) $2^{1/2}$
b.) $3^{1/2}$
c.) $ \left(\frac{3}{2}\right)^{1/2} $
d.) $ \left(\frac{1}{3}\right)^{1/2} $
My Approach: I initially tried to write the general term of the following expansion but couldn't do so due to the fact that number of components in product change with each term.
I tried using the partial fractions but couldn't proceed with it much further.
In the final attempt I tried grouping the terms successively in pairs of two, and then trying to add them , that too however proved futile.
I am looking for a method to solve the above question, as well as a more general line of approach for the following type of questions.
