# Hybrid Progression

Calculus Level 3

$\begin{eqnarray} && 3, 6,9,12,\ldots \\ && 3, 9,27,81,\ldots \\ \end{eqnarray}$

The above shows two rows of numbers, each of these rows has infinitely many numbers.
The first row of numbers follows an arithmetic progression, whereas
The second row of numbers follows a geometric progression.

We know that the sum of all the numbers in each row diverges to infinity.
However, the sum of the ratio of each term exists! What is it?

In other words, what is the value of

$\dfrac33 + \dfrac69 + \dfrac9{27} + \dfrac{12}{81} + \cdots \, ?$

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