Ch. 10 - Infinite Sequences and Series

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 10 - Infinite Sequences and Series

Problem 10.3.50

Chapter 10, Problem 10.3.50

Are there any values of x for which ∑ (from n=1 to ∞) (1 / nˣ) converges? Give reasons for your answer.

Verified step by step guidance

Recognize that the given series is the p-series (or more specifically, the Riemann zeta function series) defined as \(\sum_{n=1}^{\infty} \frac{1}{n^x}\), where \(x\) is a real number.

Recall the convergence criteria for p-series: the series \(\sum_{n=1}^{\infty} \frac{1}{n^p}\) converges if and only if \(p > 1\) and diverges otherwise.

Apply this criterion to the given series by comparing \(x\) to 1. If \(x > 1\), the terms \(\frac{1}{n^x}\) decrease sufficiently fast for the series to converge.

If \(x \leq 1\), the terms do not decrease fast enough, and the series diverges. For example, when \(x=1\), the series becomes the harmonic series, which is known to diverge.

Therefore, the series \(\sum_{n=1}^{\infty} \frac{1}{n^x}\) converges if and only if \(x > 1\).

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Key Concepts

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

Definition of the p-Series

A p-series is an infinite series of the form ∑ 1/n^p, where p is a real number. Its convergence depends on the value of p, serving as a fundamental example in series convergence tests.

Convergence Criteria for p-Series

The p-series ∑ 1/n^p converges if and only if p > 1. For p ≤ 1, the series diverges. This result is crucial for determining the convergence of series like ∑ 1/n^x by comparing the exponent x to 1.

Comparison Test for Series Convergence

The comparison test involves comparing a given series to a known benchmark series to determine convergence. If the terms of the given series are smaller than those of a convergent series, it also converges, aiding in analyzing series like ∑ 1/n^x.

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Direct Comparison Test

Related Practice

Textbook Question

Applying the Integral Test

Use the Integral Test to determine if the series in Exercises 1–12 converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.

∑ (from n = 2 to ∞) ln(n²) / n

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Textbook Question

Telescoping Series

In Exercises 39–44, find a formula for the nth partial sum of the series and use it to determine if the series converges or diverges. If a series converges, find its sum.

∑ (from n = 1 to ∞) [ (1/n) − (1/(n + 1)) ]

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Textbook Question

Intervals of Convergence

In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?

∑ (from n = 0 to ∞) (2x)ⁿ

Textbook Question

Determining Convergence or Divergence

In Exercises 1–14, determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.

∑ (from n = 2 to ∞) [(-1)ⁿ⁺¹ (1 / ln n)]

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Textbook Question

Using the Ratio Test

In Exercises 1–8, use the Ratio Test to determine whether each series converges absolutely or diverges.

∑(from n=1 to ∞) [(-1)ⁿ (n + 2) / 3ⁿ]

Textbook Question

Using the Root Test

In Exercises 9–16, use the Root Test to determine if each series converges absolutely or diverges.

∑(from n=1 to ∞) [(-1)ⁿ (1 − 1/n)ⁿ^²]

(Hint: lim (n→∞) (1 + x/n)ⁿ = eˣ)