p-series

College/University

Definition

You've got a p-series when you see that a series converges when p > 1 and diverges when p < 1. It is an infinite series in the form of \(\sum_{m=1}^{\infty } \frac{1}{m^p}\). Used in the comparison and limit comparison test.

Worked examples

\(\sum_{m=1}^{\infty} \frac{1}{m^2}\) converges because \(p = 2 > 1\)

When the exponent on m is greater than 1, the p-series converges.

\(\sum_{m=1}^{\infty} \frac{1}{m} = \sum_{m=1}^{\infty} \frac{1}{m^1}\) diverges because \(p = 1\)

The harmonic series has p = 1, so it diverges (boundary case).

\(\sum_{m=1}^{\infty} \frac{1}{\sqrt{m}} = \sum_{m=1}^{\infty} \frac{1}{m^{1/2}}\) diverges because \(p = \frac{1}{2} < 1\)

When p is less than 1, the terms don't shrink fast enough and the series diverges.

Common mistakes

\(\sum_{m=1}^{\infty} \frac{1}{m}\) converges because \(p = 1\) → \(\sum_{m=1}^{\infty} \frac{1}{m}\) diverges (\(p = 1\) is the boundary) Convergence requires p strictly greater than 1; p = 1 gives the divergent harmonic series.
\(\sum_{m=1}^{\infty} \frac{1}{2^m}\) is a p-series with \(p = 2\) → \(\sum_{m=1}^{\infty} \frac{1}{2^m}\) is geometric, not p-series A p-series has the variable m in the base raised to a constant power p, not a constant base raised to m.
\(\sum_{m=1}^{\infty} \frac{1}{m^{0.9}}\) converges because \(p = 0.9\) → \(\sum_{m=1}^{\infty} \frac{1}{m^{0.9}}\) diverges because \(p = 0.9 < 1\) Smaller p means slower decay; the series only converges when p > 1.

Where you'll use it next

You'll use p-series as a benchmark in comparison and limit comparison tests to determine convergence of more complicated series in Calculus II, and the ideas extend to integral tests and improper integrals.

Found in 1 StudyPug lesson

P Series: Exploring Convergence and Divergence in Infinite Sums

Calculus 2

Dive into the fascinating world of P Series. Understand convergence criteria, explore divergence cases, and discover real-world applications in physics and engineering. Elevate your calculus skills today!

UniversityUniversity