The sum 
\[ \sum_{k=0}^{\infty} \frac{2^{k}}{5^{2^{k}}+1}\]
can be written in the form $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
1. **Identify the series and the goal**: We need to find the sum 
   \[
   \sum_{k=0}^{\infty} \frac{2^{k}}{5^{2^{k}}+1}
   \]
   and express it in the form \(\frac{p}{q}\) where \(p\) and \(q\) are relatively prime positive integers, and then find \(p+q\).

2. **Use the identity for the series**: The given solution uses the identity 
   \[
   \sum_{k=0}^\infty \frac{1}{N+1} = \sum_{k=0}^\infty \frac{1/N}{1-(-1/N)} = \frac{1}{N} - \frac{1}{N^2} + \frac{1}{N^3} - \cdots
   \]
   for \(N > 1\). This identity is useful for transforming the series into a more manageable form.

3. **Transform the given series**: We apply the identity to our series:
   \[
   \sum_{k=0}^\infty \frac{2^k}{5^{2^k}+1}
   \]
   Using the identity, we rewrite each term:
   \[
   \frac{2^k}{5^{2^k}+1} = \frac{2^k}{5^{2^k}} \cdot \frac{1}{1 + \frac{1}{5^{2^k}}} = \frac{2^k}{5^{2^k}} \left(1 - \frac{1}{5^{2^k}} + \frac{1}{5^{2 \cdot 2^k}} - \cdots \right)
   \]
   This simplifies to:
   \[
   \sum_{k=0}^\infty \frac{2^k}{5^{2^k}} - \sum_{k=0}^\infty \frac{2^k}{5^{2 \cdot 2^k}} + \sum_{k=0}^\infty \frac{2^k}{5^{3 \cdot 2^k}} - \cdots
   \]

4. **Rearrange the series**: We can rearrange the series by summing over all powers of 5:
   \[
   \sum_{n \geq 1} \left( \frac{2^{\nu_2(n)}}{5^n} - \sum_{i=0}^{\nu_2(n)-1} \frac{2^i}{5^n} \right)
   \]
   Here, \(\nu_2(n)\) denotes the highest power of 2 dividing \(n\).

5. **Simplify the series**: Notice that the inner sum cancels out many terms, leaving us with:
   \[
   \sum_{n \geq 1} \frac{1}{5^n}
   \]
   This is a geometric series with the first term \(a = \frac{1}{5}\) and common ratio \(r = \frac{1}{5}\).

6. **Sum the geometric series**: The sum of an infinite geometric series is given by:
   \[
   \sum_{n=1}^\infty \frac{1}{5^n} = \frac{\frac{1}{5}}{1 - \frac{1}{5}} = \frac{\frac{1}{5}}{\frac{4}{5}} = \frac{1}{4}
   \]

7. **Conclusion**: The sum of the series is \(\frac{1}{4}\). Therefore, \(p = 1\) and \(q = 4\), and \(p+q = 1+4 = 5\).

The final answer is \(\boxed{5}\)