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Are there any values of x for which ∑ (from n=1 to ∞) (1 / nˣ) converges? Give reasons for your answer.
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Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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
Capitolo 10, Problema 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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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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Arithmetic Sequences - Recursive Formula
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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8:22Introduction to Sequences
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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6:40Arithmetic Sequences - Recursive Formula
Pratica correlata
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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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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]
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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)
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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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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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