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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblem 10.3.60b
Chapter 10, Problem 10.3.60b

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.

Verified step by step guidance
1
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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Expand & Condense Log Expressions

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.
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P-Series and Harmonic 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.
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Geometric Series
Related Practice
Textbook Question

∑ (from n=1 to ∞) (1 / √(n + 1)) diverges

b. What should n be in order that the partial sum sₙ = ∑ (from i=1 to n) (1 / √(i + 1)) satisfies sₙ > 1000?

Textbook Question

(Continuation of Exercise 61.) Use the result in Exercise 61 to determine which of the following series converge and which diverge. Support your answer in each case.

a. ∑ (from n=2 to ∞) [1 / (n ln n)]

Textbook Question

b. From Example 5, Section 10.2, show that

S = 1 + ∑(from n=1 to ∞) [1 / (n²(n + 1))].

Textbook Question

The series

sec x = 1 + x²/2 + 5x⁴/24 + 61x⁶/720 + 277x⁸/8064 + ⋯

converges to sec x for −π/2 < x < π/2.

a. Find the first five terms of a power series for the function ln|sec x + tan x|. For what values of x should the series converge?

Textbook Question

Intervals of Convergence

In Exercises 1–36, for what values of x does the series converge (b) absolutely?

∑ (from n = 1 to ∞) [ (√(n + 1) − √n)(x − 3)ⁿ ]

Textbook Question

Intervals of Convergence

In Exercises 1–36, for what values of x does the series converge (b) absolutely?

∑ (from n = 0 to ∞) [ (−2)ⁿ (n + 1) (x − 1)ⁿ ]