sandrareed2005
sandrareed2005 5d ago • 0 views

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

Hey there! 👋 Struggling with p-series convergence and divergence? Don't worry, I've got you covered! This guide breaks down the steps and includes a quiz to test your knowledge. Let's ace this together! 💯
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samuel.gay Dec 27, 2025

📚 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.
Click to see Answers
  1. C
  2. C
  3. B
  4. D
  5. B
  6. A
  7. C

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