p-series

Definition

A p-series is a specific type of benchmark infinite series of the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$.

• How to read: “The sum from n equals one to infinity of one divided by n raised to the p.”
• Meaning: It is the standard comparison series for convergence tests, and its behavior depends entirely on the exponent $p$.

Why It Matters

The p-series acts as the “measuring stick” for convergence of other series. By understanding when a p-series converges or diverges, we can quickly evaluate more complex systems through comparison tests.

Core Concepts

• Convergence Criteria:
  • Converges if $p > 1$.
    • How to read: “The value p is greater than one.”
    • Meaning: Terms shrink fast enough (like $1/n^2$) for the total to stay finite.
  • Diverges if $p \leq 1$.
    • How to read: “The value p is less than or equal to one.”
    • Meaning: Terms decay too slowly; the partial sums grow without bound.
• Harmonic Series: The case $p=1$ ($\sum 1/n$) is the most famous divergent series, where the terms go to zero but the sum still approaches infinity.
  • How to read: “The case where p is equal to one, representing the sum of one divided by n.”
  • Meaning: Terms vanish individually yet the cumulative sum diverges—classic example that $\lim a_n = 0$ does not imply convergence.

Connected Concepts

• The Integral Test
• The nth-Term Test