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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Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
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)
Verified step by step guidance1
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
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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Textbook Question
In Exercises 81–86, find the values of x for which the given geometric series converges.
Also, find the sum of the series (as a function of x) for those values of x.
∑ (from n = 0 to ∞) [ (−1/2)ⁿ (x − 3)ⁿ ]
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!)