Steps to identify and test p-Series for convergence or divergence.

📚 Quick Study Guide

• 🔍 A p-series is a series of the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$, where $p$ is a positive real number.
• 🔢 The convergence or divergence of a p-series depends solely on the value of $p$.
• ✅ If $p > 1$, the p-series converges. This is known as the p-series test.
• ❌ If $p \le 1$, the p-series diverges. This includes the special case where $p = 1$, which is the harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$.
• ➕ To identify if a series is a p-series, check if it can be written in the form $\frac{1}{n^p}$.
• ➗ To test for convergence or divergence, determine the value of $p$ and compare it to 1.
• 💡 Remember that the p-series test only applies to series that are exactly in the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$. If the series is more complex, other convergence tests may be needed.

Practice Quiz

1. Which of the following series is a p-series?

   1. $\sum_{n=1}^{\infty} \frac{1}{n^2 + 1}$
   2. $\sum_{n=1}^{\infty} \frac{1}{2^n}$
   3. $\sum_{n=1}^{\infty} \frac{1}{n}$
   4. $\sum_{n=1}^{\infty} \frac{n}{n+1}$

2. For what values of $p$ does the series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converge?

   1. $p < 1$
   2. $p \le 1$
   3. $p > 1$
   4. $p \ge 1$

3. Does the series $\sum_{n=1}^{\infty} \frac{1}{n^{0.5}}$ converge or diverge?

   1. Converges
   2. Diverges
   3. Cannot be determined
   4. Converges conditionally

4. Which of the following p-series converges?

   1. $\sum_{n=1}^{\infty} \frac{1}{n}$
   2. $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$
   3. $\sum_{n=1}^{\infty} \frac{1}{n^{0.99}}$
   4. $\sum_{n=1}^{\infty} \frac{1}{n^{1.01}}$

5. What is the value of $p$ for the series $\sum_{n=1}^{\infty} \frac{1}{n^3}$?

   1. 1
   2. 3
   3. -3
   4. 1/3

6. Does the series $\sum_{n=1}^{\infty} \frac{5}{n^2}$ converge or diverge?

   1. Converges
   2. Diverges
   3. Cannot be determined
   4. Converges conditionally

7. Which statement is correct about the harmonic series?

   1. It is a p-series with p > 1 and converges.
   2. It is a p-series with p = 1 and converges.
   3. It is a p-series with p = 1 and diverges.
   4. It is not a p-series.