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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 10.1.105
Chapter 10, Problem 10.1.105

Recursively Defined Sequences
In Exercises 101–108, assume that each sequence converges and find its limit.
a₁ = 5,aₙ₊₁ = √(5aₙ)

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Identify the recursive sequence given: \(a_1 = 5\) and \(a_{n+1} = \sqrt{5a_n}\). 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 \(L = \sqrt{5L}\) to find the limit.
Square both sides of the equation to eliminate the square root: \(L^2 = 5L\).
Rearrange the equation to standard polynomial form: \(L^2 - 5L = 0\).
Factor the equation: \(L(L - 5) = 0\). The possible limits are \(L = 0\) or \(L = 5\). Use the initial term and the nature of the sequence to determine which limit is valid.

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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 based on one or more previous terms. Understanding how to work with such sequences involves using the given formula repeatedly to generate terms and analyze their behavior as n increases.
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Convergence of Sequences

A sequence converges if its terms approach a specific finite value as n approaches infinity. Determining convergence often involves finding a limit L such that the sequence terms get arbitrarily close to L, which is essential for solving limit problems in recursive sequences.
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Finding Limits of Recursive Sequences

To find the limit of a convergent recursive sequence, assume the limit exists and set L equal to the limit of both aₙ and aₙ₊₁. Then solve the resulting equation, often involving algebraic manipulation, to find the value of L that satisfies the recursive definition.
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Related Practice
Textbook Question

Direct Comparison Test

In Exercises 1–8, use the Direct Comparison Test to determine if each series converges or diverges.

∑ (from n=1 to ∞) (√n + 1) / (√(n² + 3))

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Textbook Question

In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.

∑ (from n = 2 to ∞) [(ln n / n)³]

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 ∞) sin (1/n)

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Textbook Question

In Exercises 37–42, find the series’ radius of convergence.

∑ (from n = 1 to ∞) [ (n!)² / (2ⁿ (2n)!) ] xⁿ

Textbook Question

Finding nth Partial Sums

In Exercises 1–6, find a formula for the nth partial sum of each series and use it to find the series’ sum if the series converges.

2 + (2/3) + (2/9) + (2/27) + … + (2 / 3ⁿ⁻¹) + …

Textbook Question

Finding a Sequence’s Formula

In Exercises 13–30, find a formula for the nth term of the sequence.

2, 6, 10, 14, 18, …Every other even positive integer

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