Find the sum of the following series to infinity: `2//5+3//5^2\ +2//5^3+3//5^4+oo`

Explanation

To find the sum of the series \( S = \frac{2}{5} + \frac{3}{5^2} + \frac{2}{5^3} + \frac{3}{5^4} + \ldots \) to infinity, we can observe that the series alternates between two types of terms. Let's break it down step by step.

### Step 1: Identify the pattern in the series
The series consists of two types of terms:
- Odd indexed terms: \( \frac{2}{5}, \frac{2}{5^3}, \frac{2}{5^5}, \ldots \)
- Even indexed terms: \( \frac{3}{5^2}, \frac{3}{5^4}, \frac{3}{5^6}, \ldots \)

### Step 2: Split the series into two separate series
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