I have to study find the closing form of the following series:
\begin{equation} f(1) + f(2) + f(3) + f(5) + f(6) + f(7) + \dots \end{equation}
So basically the sum of all terms without the multiple of 4. Each term of the series is defined as: $f(i) = p^i$ with $0 < p < 1$ that is a constant. Rewriting the previous equation we obtain:
\begin{equation} p^1 + p^2 + p^3 + p^5 + p^6 + p^7 + \dots \end{equation}
So baiscally this is a geometric series without the multiple of 4.
My idea to handle this was to divide the series in 3 different subseries
\begin{equation} p^1 + p^5 + p^9 + \dots + p^2 + p^6 + p^{10} + \dots + p^3 + p^7 + p^{11} + \dots \end{equation}
\begin{equation} \sum_{i=1}^{+\infty}p^{4i - 3} + \sum_{i=1}^{+\infty}p^{4i - 2} + \sum_{i=1}^{+\infty}p^{4i - 1} \end{equation}
In this way each of the subseries can be solved easily with a change of variable ($p^{4} = q)$.
My doubt is that since the series is an infinite series, this grouping cannot be done.
Is my solution correct? Or there is another way to handle this kind of series?
Given that all the terms of the series are positive, rearranging (regrouping) them in any way us legitimate, so what you did is completely correct.
The only situation when rearranging terms of a series might change the value of the sum is when the series contains both positive numbers and negative numbers, and the sum of all the positive numbers is infinite and so is the sum of all the negative numbers.