A p-series is a type of infinite series where the general term of the series is given by $\frac{1}{n^p}$, where $p$ is a real number. The convergence or divergence of a p-series is determined by the value of $p$, which is a crucial concept in the context of the Divergence and Integral Tests, Comparison Tests, and Ratio and Root Tests.
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The p-series converges if $p > 1$ and diverges if $p \leq 1$.
The Divergence Test states that if the general term of a series satisfies $\lim_{n\to\infty} a_n = 0$, then the series may converge or diverge.
The Integral Test can be used to determine the convergence or divergence of a p-series by comparing the series to the improper integral $\int_1^\infty \frac{1}{x^p} dx$.
The Comparison Test allows us to compare a p-series to another series with known convergence or divergence properties to determine the behavior of the p-series.
The Ratio Test and Root Test can also be used to analyze the convergence or divergence of a p-series by examining the behavior of the general term.
Review Questions
Explain how the value of $p$ in a p-series determines the convergence or divergence of the series.
The value of $p$ in a p-series, $\sum_{n=1}^{\infty} \frac{1}{n^p}$, is the key factor in determining the convergence or divergence of the series. If $p > 1$, the series converges, meaning the sum of the series approaches a finite value as the number of terms increases without bound. If $p \leq 1$, the series diverges, meaning the sum of the series does not approach a finite value as the number of terms increases without bound.
Describe how the Divergence Test and Integral Test can be used to analyze the convergence or divergence of a p-series.
The Divergence Test states that if the general term of a series satisfies $\lim_{n\to\infty} a_n = 0$, then the series may converge or diverge. For a p-series, the general term is $\frac{1}{n^p}$, which satisfies this condition. The Integral Test can then be used to determine the convergence or divergence of the p-series by comparing the series to the improper integral $\int_1^\infty \frac{1}{x^p} dx$. If the integral converges, the p-series converges; if the integral diverges, the p-series diverges.
Analyze how the Comparison Test, Ratio Test, and Root Test can be used to determine the convergence or divergence of a p-series.
The Comparison Test allows us to compare a p-series to another series with known convergence or divergence properties, such as a geometric series, to determine the behavior of the p-series. The Ratio Test and Root Test can also be used to analyze the convergence or divergence of a p-series by examining the behavior of the general term, $\frac{1}{n^p}$. If the limit of the ratio or root of the general term is less than 1, the series converges; if the limit is greater than or equal to 1, the series diverges. These tests provide additional tools for determining the convergence or divergence of a p-series.
An infinite series is the sum of an infinite number of terms, typically represented as $\sum_{n=1}^{\infty} a_n$, where $a_n$ is the general term of the series.
Convergence refers to the property of an infinite series where the sum of the series approaches a finite value as the number of terms increases without bound.
Divergence refers to the property of an infinite series where the sum of the series does not approach a finite value as the number of terms increases without bound.