Find the sum of the following series to infinity: `1/3+1/(5^2)+1/(3^3)+1/(5^4)+1/(3^5)+1/(5^6)+oo`

Explanation

To find the sum of the series \( S = \frac{1}{3} + \frac{1}{5^2} + \frac{1}{3^3} + \frac{1}{5^4} + \frac{1}{3^5} + \frac{1}{5^6} + \ldots \) to infinity, we can separate the series into two parts: the odd terms and the even terms.

### Step 1: Identify the odd and even terms
The odd terms of the series are:
\[ S_{\text{odd}} = \frac{1}{3} + \frac{1}{3^3} + \frac{1}{3^5} + \ldots \]
The even terms of the series are:
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