For each given p -series, identify p and determine whether...

For each given $p$ -series, identify $p$ and determine whether the series converges. $$ \text { (a) } \sum_{k=1}^{\infty} k^{-4 / 3} \text { (b) } \sum_{k=1}^{\infty} \frac{1}{\sqrt[4]{k}} \quad \text { (c) } \sum_{k=1}^{\infty} \frac{1}{\sqrt[3]{k^{5}}} \quad \text { (d) } \sum_{k=1}^{\infty} \frac{1}{k^{\pi}} $$

For each given $p$ -series, identify $p$ and determine whether the series converges.
$$
\text { (a) } \sum_{k=1}^{\infty} k^{-4 / 3} \text { (b) } \sum_{k=1}^{\infty} \frac{1}{\sqrt[4]{k}} \quad \text { (c) } \sum_{k=1}^{\infty} \frac{1}{\sqrt[3]{k^{5}}} \quad \text { (d) } \sum_{k=1}^{\infty} \frac{1}{k^{\pi}}
$$

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Key Concepts

p-Series

A p-series is an infinite series of the form ? (1/k^p), where k is a positive integer and p is a constant exponent. It is a fundamental series encountered in calculus and analysis, and its convergence properties depend solely on the value of the exponent p. Understanding p-series is crucial because they serve as a benchmark for comparing the behavior of more complex series.

Convergence Criterion for p-Series

The convergence of a p-series is determined by the inequality on the exponent p: the series converges if and only if p > 1, and it diverges when p ? 1. This simple test – sometimes called the p-test – is a powerful criterion used in evaluating the convergence of series and is one of the first convergence tests taught in calculus.

Manipulating Exponents and Roots

In many series, the terms may be expressed using roots instead of exponentials. Converting these roots to fractional exponent form allows one to identify the effective exponent p and apply the p-series test. Mastery of exponent manipulation is therefore essential for analyzing the convergence or divergence of series presented in various forms.

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