Problem 83

Question

Define a $p$ -series and state the requirements for its convergence.

Step-by-Step Solution

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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 $p \leq 1$.

1Step 1: 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.

2Step 2: 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.

Problem 83

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Problem 83

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Problem 84

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