To determine the value of $p$ so that the series

$\sum_{n=1}^{\infty} \frac{n^p}{n^{p+5}}$

converges, we can simplify the expression and apply the convergence criteria for a series.

Solution Steps:

1. Simplify the Term: The given term is: $\frac{n^p}{n^{p+5}} = n^{p - (p+5)} = n^{-5}$

   This simplifies the series to: $\sum_{n=1}^{\infty} \frac{1}{n^5}$

2. Analyze the Simplified Series: The series now looks like a $p$-series, which takes the form: $\sum_{n=1}^{\infty} \frac{1}{n^q}$

   A $p$-series converges if $q > 1$. In our case, we have $q = 5$, which is greater than 1. Therefore, the series converges for any $p$ because the power of $n$ in the denominator is sufficient to ensure convergence.

Conclusion:

The series converges for any value of $p$ since the simplification led to a convergent $p$-series with $q = 5$.

Would you like a deeper explanation of each step or have any questions?

Here are some related questions that expand on this topic:

1. What are $p$-series, and how is their convergence determined?
2. Can you explain the Ratio Test and how it applies to series convergence?
3. How does the comparison test work for determining series convergence?
4. What happens if $p+5$ in the denominator changes to another function of $n$?
5. How would convergence criteria change if we had an additional constant term in the numerator?

Tip: Understanding series convergence criteria like the $p$-series test, Ratio Test, and Comparison Test can greatly help in quickly determining convergence for various types of series.