The given mathematical expression is:

$X^p = \sum_{i} \frac{1}{1 + i + p_i} \quad \text{for} \quad (\sum_{i=1}^{\infty} \frac{1}{p_i})$

Here's a breakdown of the elements in the equation:

1. Left Side ( $X^p$ ): This appears to be the main variable or expression in terms of the variables $X$ and $p$.
2. Summation Notation ( $\sum_{i}$ ): This indicates a summation over the index $i$.
3. Fraction Expression: The term inside the summation is given as $\frac{1}{1 + i + p_i}$. Here:
   • $i$ is the index variable.
   • $p_i$ could represent a series of terms depending on $i$.
4. Condition for Summation ($\sum_{i=1}^{\infty} \frac{1}{p_i}$): This indicates that the value of $X^p$ may depend on an infinite series of the form $\sum_{i=1}^{\infty} \frac{1}{p_i}$.

To analyze and solve or further explore the expression, more context might be needed about $p_i$ and its properties.

Follow-up Questions:

1. Do you want to solve this expression for a specific value of $p$?
2. Should we explore the convergence of the series $\sum_{i=1}^{\infty} \frac{1}{p_i}$?
3. Are there constraints or properties for $p_i$ that should be considered?
4. Would you like a detailed explanation of the convergence criteria?
5. Should I help with simplifying the given summation?

Tip:

When dealing with series like $\sum_{i=1}^{\infty} \frac{1}{p_i}$, it's essential to analyze their convergence using tests like the ratio test or integral test.