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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 10.1.101
Chapter 10, Problem 10.1.101

Recursively Defined Sequences
In Exercises 101–108, assume that each sequence converges and find its limit.
a₁ = 2,aₙ₊₁ = 72 / (1 + aₙ)

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Identify the recursive sequence given: \(a_1 = 2\) and \(a_{n+1} = \frac{72}{1 + a_n}\). We want to find the limit \(L\) as \(n\) approaches infinity, assuming the sequence converges.
Assuming the sequence converges to a limit \(L\), then both \(a_n\) and \(a_{n+1}\) approach \(L\). So, set \(L = \frac{72}{1 + L}\).
Multiply both sides of the equation by \((1 + L)\) to clear the denominator: \(L(1 + L) = 72\).
Rewrite the equation as a quadratic: \(L^2 + L - 72 = 0\).
Solve the quadratic equation for \(L\) using the quadratic formula \(L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=1\), \(b=1\), and \(c=-72\). Then determine which root makes sense in the context of the sequence.

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Key Concepts

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

Recursively Defined Sequences

A recursively defined sequence is one where each term is defined in terms of one or more previous terms. Understanding how to use the given initial term and the recursive formula is essential to generate terms and analyze the sequence's behavior.
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Convergence of Sequences

A sequence converges if its terms approach a specific finite value as the index goes to infinity. Determining convergence involves analyzing the recursive formula and checking if the sequence settles to a stable limit.
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Finding the Limit of a Recursive Sequence

To find the limit of a convergent recursive sequence, assume the limit exists and set it equal to the expression defining the next term. Solving the resulting equation yields the limit, which represents the value the sequence approaches.
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Related Practice
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)

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 = nπ cos(nπ)

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Textbook Question

Uniqueness of limits Prove that limits of sequences are unique. That is, show that if L₁ and L₂ are numbers such that aₙ → L₁ and aₙ → L₂, then L₁ = L₂.

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Textbook Question

Use power series operations to find the Taylor series at x = 0 for the functions in Exercises 13–30.

sin x – x + (x³ / 3!)