Use the Cauchy condensation test from Exercise 59 to show that:
b. ∑ (from n=1 to ∞) [1 / nᵖ] converges if p > 1 and diverges if p ≤ 1.

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Recall the Cauchy condensation test: For a non-increasing sequence of positive terms \(a_n\), the series \(\sum_{n=1}^\infty a_n\) converges if and only if the series \(\sum_{k=0}^\infty 2^k a_{2^k}\) converges.

Apply the test to the series \(\sum_{n=1}^\infty \frac{1}{n^p}\) by setting \(a_n = \frac{1}{n^p}\), which is positive and decreasing for \(p > 0\).

Form the condensed series: \(\sum_{k=0}^\infty 2^k a_{2^k} = \sum_{k=0}^\infty 2^k \cdot \frac{1}{(2^k)^p} = \sum_{k=0}^\infty 2^k \cdot 2^{-kp} = \sum_{k=0}^\infty 2^{k(1-p)}\).

Analyze the convergence of the geometric series \(\sum_{k=0}^\infty 2^{k(1-p)}\): it converges if and only if the common ratio \(r = 2^{1-p}\) satisfies \(|r| < 1\), which means \(1 - p < 0\) or \(p > 1\).

Conclude that by the Cauchy condensation test, the original series \(\sum_{n=1}^\infty \frac{1}{n^p}\) converges if \(p > 1\) and diverges if \(p \leq 1\).

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주요 개념

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Cauchy Condensation Test

The Cauchy condensation test is a method to determine the convergence of series with positive, non-increasing terms. It states that the series ∑a_n converges if and only if the condensed series ∑2^n a_{2^n} converges. This test simplifies analyzing series by focusing on terms at powers of two.

p-Series and Their Convergence

A p-series is a series of the form ∑1/n^p, where p is a real number. Its convergence depends on the value of p: the series converges if p > 1 and diverges if p ≤ 1. This result is fundamental in understanding the behavior of many infinite series.

Comparison of Series Using Condensation

Applying the Cauchy condensation test to the p-series transforms it into a simpler geometric series ∑2^{n(1-p)}. Analyzing this geometric series helps determine convergence by comparing the exponent's sign, linking the original series' behavior to the parameter p.

Use the Cauchy condensation test from Exercise 59 to show that:b. ∑ (from n=1 to ∞) [1 / nᵖ] converges if p > 1 and diverges if p ≤ 1.