The given series is:

$\sum_{n=1}^{\infty} \frac{n^p}{n^q} = \sum_{n=1}^{\infty} n^{p-q}$

where $p$ and $q$ are positive numbers.

Step 1: Simplifying the Series

The general term of the series simplifies to:

$n^{p-q}$

So the series becomes:

$\sum_{n=1}^{\infty} n^{p-q}$

Step 2: Applying the p-series Test

This series resembles a p-series, which is of the form:

$\sum_{n=1}^{\infty} \frac{1}{n^r}$

for some value of $r$. In our case, the term is $n^{p-q} = n^{-(q-p)}$. Thus, we can compare this to a p-series with exponent $-(q-p)$.

The p-series convergence criterion states:

• If $r > 1$, the series converges.
• If $r \leq 1$, the series diverges.

Step 3: Testing Convergence

For our series to converge, we need the exponent $q - p > 1$, meaning that $q > p + 1$. If this condition holds, the series converges. Otherwise, the series diverges.

Conclusion:

• If $q > p + 1$, the series converges.
• If $q \leq p + 1$, the series diverges.

Would you like a deeper explanation on any step? Here are five related questions to consider:

1. How does the p-series test work for other series forms?
2. Can this series be tested using the ratio test for more complex cases?
3. What happens if $p = q$ or $p > q$?
4. How would the behavior of the series change if we altered the powers of $n$?
5. Can we express the sum of the series in closed form for specific values of $p$ and $q$?

Tip: Understanding convergence tests (like p-series and ratio tests) is essential for working with infinite series.