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Estimate the value of ∑ (from n=2 to ∞) (1 / (n² + 4)) to within 0.1 of its exact value.
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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 10.1.22
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
Guida verificata passo dopo passo1
Identify the pattern in the sequence: 2, 6, 10, 14, 18, ... Notice that each term increases by a constant difference of 4, which means this is an arithmetic sequence.
Recall the general formula for the nth term of an arithmetic sequence: \(a_n = a_1 + (n - 1)d\), where \(a_1\) is the first term and \(d\) is the common difference.
Substitute the known values into the formula: the first term \(a_1 = 2\) and the common difference \(d = 4\), so the formula becomes \(a_n = 2 + (n - 1) \times 4\).
Simplify the expression by distributing the 4: \(a_n = 2 + 4n - 4\).
Combine like terms to write the formula in simplest form: \(a_n = 4n - 2\).
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Arithmetic Sequences
An arithmetic sequence is a list of numbers where each term after the first is found by adding a constant difference. Recognizing this pattern helps in formulating the nth term by identifying the first term and the common difference.
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5:17
Arithmetic Sequences - General Formula
General Formula for the nth Term
The nth term of an arithmetic sequence can be expressed as a_n = a_1 + (n - 1)d, where a_1 is the first term and d is the common difference. This formula allows direct calculation of any term in the sequence.
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Percorso guidato
05:44Divergence Test (nth Term Test)
Sequence Pattern Recognition
Identifying the pattern in the given sequence, such as recognizing it lists every other even positive integer, is essential. This insight guides the choice of the first term and common difference in the formula.
Video consigliato:
8:22Introduction to Sequences
Pratica correlata
Domanda del libro di testo
In Exercises 37–42, find the series’ radius of convergence.
∑ (from n = 1 to ∞) [ (n!)² / (2ⁿ (2n)!) ] xⁿ
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Domanda del libro di testo
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) + …
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Domanda del libro di testo
Absolute and Conditional Convergence
Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ / (1 + √n)]
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Domanda del libro di testo
Finding Taylor Series at x = 0 (Maclaurin Series)
Find the Maclaurin series for the functions in Exercises 11–24.
5 cos πx
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Domanda del libro di testo
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 ∞) (2ⁿ + 3ⁿ) / (3ⁿ + 4ⁿ)
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