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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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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 10.1.4
Finding Terms of a Sequence
Each of Exercises 1–6 gives a formula for the nth term aₙ of a sequence {aₙ}. Find the values of a₁, a₂, a₃, and a₄.
aₙ = 2 + (-1)ⁿ
Guida verificata passo dopo passo1
Identify the given formula for the nth term of the sequence: \(a_{n} = 2 + (-1)^{n}\).
Recall that to find specific terms of the sequence, substitute the term number \(n\) into the formula.
Calculate \(a_{1}\) by substituting \(n=1\) into the formula: \(a_{1} = 2 + (-1)^{1}\).
Calculate \(a_{2}\) by substituting \(n=2\) into the formula: \(a_{2} = 2 + (-1)^{2}\).
Similarly, find \(a_{3}\) and \(a_{4}\) by substituting \(n=3\) and \(n=4\) respectively into the formula: \(a_{3} = 2 + (-1)^{3}\) and \(a_{4} = 2 + (-1)^{4}\).
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Sequences and Terms
A sequence is an ordered list of numbers defined by a specific rule or formula for its terms. Each term is identified by its position n, and the nth term aₙ gives the value at that position. Understanding how to interpret and use the formula for aₙ is essential to find specific terms.
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8:22
Introduction to Sequences
Substitution in Formulas
To find specific terms of a sequence, substitute the term number n into the given formula. This involves replacing n with 1, 2, 3, etc., and simplifying the expression to calculate the corresponding term values accurately.
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04:27Substitution With an Extra Variable
Properties of Exponents and Alternating Signs
The term (-1)ⁿ alternates between -1 and 1 depending on whether n is odd or even. Recognizing this pattern helps determine how the sign affects each term in the sequence, which is crucial for correctly evaluating the formula.
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7:39Introduction to Exponent Rules
Pratica correlata
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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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Series with Geometric Terms
In Exercises 7–14, write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges.
∑ (from n = 0 to ∞) [(1 / 2ⁿ) + ((-1)ⁿ / 5ⁿ)]
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Determining Convergence or Divergence
Which of the series in Exercises 13–46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)
∑ (from n=1 to ∞) (1 + 1/n)ⁿ
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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√10)]
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Suppose that aₙ > 0 and limₙ→∞ n²aₙ = 0. Prove that ∑aₙ converges.
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