Question

Define a \(p\) -series and state the requirements for its convergence.

Short answer

A \(p\)-series is defined as the series \(\sum_{n=1}^{\infty} \frac{1}{n^p}\), for \(n\) a natural number and \(p\) a real number. The series is convergent if \(p > 1\), and divergent if \(p \leq 1\).

Step-by-step solution

Define a \(p\)-series

A \(p\)-series can be defined as the series of the form \(\sum_{n=1}^{\infty} \frac{1}{n^p}\), where \(n\) is a natural number and \(p\) is a real number. This series derives its name from the power \(p\) of \(n\) in the denominator.

State the requirements for its convergence

The \(p\)-series \(\sum_{n=1}^{\infty} \frac{1}{n^p}\) is convergent if \(p > 1\) and divergent if \(p \leq 1\). This is a conclusion derived from the p-series test (part of the integral test for convergence), a fundamental concept in the study of calculus and analysis.