Determine whether the series is convergent or divergent.

1 + 

1

8

 + 

1

27

 + 

1

64

 + 

1

125

 +   

The given series is:

\[
1 + \frac{1}{8} + \frac{1}{27} + \frac{1}{64} + \frac{1}{125} + \cdots
\]

Notice that the series terms can be rewritten as:

\[
\sum_{n=1}^{\infty} \frac{1}{n^3}
\]

This is a p-series with the general form:

\[
\sum_{n=1}^{\infty} \frac{1}{n^p}
\]

In this case, \( p = 3 \).

### Convergence of p-Series:
A p-series converges if \( p > 1 \). Since \( p = 3 \), which is greater than 1, the series converges.

**Answer:** The series is **convergent**.

Would you like more details or have any questions? Here are some related questions for further exploration:

1. What is the convergence criterion for a p-series?
2. How would the series behave if \( p \) were less than 1?
3. What is the significance of the exponent in determining the convergence of a series?
4. Can you apply the integral test to confirm the convergence of this series?
5. What are other common examples of convergent series?

**Tip:** Always look for patterns in the terms of a series that can help you apply known convergence tests, like the p-series or geometric series tests.

Learn how to determine the convergence of the series 1 + 1/8 + 1/27 + 1/64 + 1/125 + ..., which is equivalent to a p-series with p = 3. Understand the conditions under which a p-series converges, using the convergence of p-series theorem. Suitable for college-level mathematics students.

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