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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 10.6.66
Chapter 10, Problem 10.6.66

In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑ (from n = 0 to ∞) [((n + 1) / (n + 2))ⁿ]

Verified step by step guidance
1
Identify the general term of the series: \(a_n = \left( \frac{n+1}{n+2} \right)^n\).
To determine convergence or divergence, consider using the Root Test or the Ratio Test. The Root Test is often simpler for terms raised to the power \(n\).
Apply the Root Test by evaluating \(\lim_{n \to \infty} \sqrt[n]{|a_n|} = \lim_{n \to \infty} \left| \frac{n+1}{n+2} \right|\).
Simplify the limit: since \(\frac{n+1}{n+2} = 1 - \frac{1}{n+2}\), the limit approaches 1 as \(n\) approaches infinity.
Interpret the Root Test result: if the limit equals 1, the test is inconclusive, so consider using the Ratio Test or analyze the behavior of \(a_n\) directly to decide on convergence or divergence.

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Key Concepts

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

Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. Determining whether a series converges means checking if the sum approaches a finite limit as the number of terms grows indefinitely. Understanding convergence is essential to analyze the behavior of the given series.
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Convergence of an Infinite Series

Limit of the General Term

A necessary condition for a series to converge is that its general term approaches zero as n approaches infinity. If the limit of the term does not equal zero, the series diverges. Evaluating this limit helps quickly assess the possibility of convergence.
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One-Sided Limits

Root Test for Series Convergence

The root test uses the nth root of the absolute value of the general term to determine convergence. If the limit of this root is less than one, the series converges absolutely; if greater than one, it diverges. This test is particularly useful for series with terms raised to the nth power.
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Root Test
Related Practice
Textbook Question

Convergence and Divergence

Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.

aₙ = (1/n) ∫₁ⁿ (1/x) dx

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Textbook Question

Which series in Exercises 53–76 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.

∑ (from n = 0 to ∞) e^(−2n)

Textbook Question

Does the series

∑ (from n=1 to ∞) (1/n − 1/n²)

converge or diverge? Justify your answer.

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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 = 1 to ∞) [ (x − 1)ⁿ / √n ]

Textbook Question

Determining Convergence or Divergence

Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.

∑ (from n=1 to ∞) (n / (3n + 1))ⁿ

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Textbook Question

Limit Comparison Test

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

∑ (from n=1 to ∞) n(n + 1) / ((n² + 1)(n − 1))

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