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Harmonic Series and p-Series

Benjamin Wright

Benjamin Wright

5 min read

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Study Guide Overview

This study guide covers p-series in calculus, including their definition and the rules for convergence and divergence. It provides practice problems to determine convergence or divergence based on the value of p. The harmonic series is introduced as a special case of a p-series. Examples demonstrate simplification techniques for applying the p-series test.

10.5 Harmonic Series and p-Series

Welcome back to Calc BC! As you keep chugging on Unit 10, you’ll find that there’s more to number series (and the course as a whole) than the geometric series we defined in 10.2 Working with Geometric Series:

🧠 P-Series in Calculus

Before we define them, let’s take a look at the following series of numbers:

1+18+127+164+...1+\frac{1}{8}+\frac{1}{27}+\frac{1}{64}+...

What do the terms have in common? If you look at the denominator, you’ll notice that it is the cube (multiply a number by itself three times).

1+\frac{1}{8}+\frac{1}{27}+\frac{1}{64}+...=\frac{1}{1^3}+\frac{1}{2^3}+...
Previous Topic - Integral Test for ConvergenceNext Topic - Comparison Tests for Convergence

Question 1 of 7

Which of the following series is a p-series? 🤔

∑n=1∞12n\sum_{n=1}^{\infty} \frac{1}{2^n}

∑n=1∞nn2+1\sum_{n=1}^{\infty} \frac{n}{n^2+1}

∑n=1∞1n3\sum_{n=1}^{\infty} \frac{1}{n^{3}}

∑n=1∞(−1)nn\sum_{n=1}^{\infty} \frac{(-1)^n}{n}