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If ∑aₙ is a convergent series of positive terms, prove that ∑sin(aₙ) converges.
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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 10.7.46
In Exercises 43–50, use Theorem 20 to find the series’ interval of convergence and, within this interval, the sum of the series as a function of x.
∑ (from n = 0 to ∞) [(x + 1)²ⁿ] / 9ⁿ
Guida verificata passo dopo passo1
Recognize that the given series is a geometric series of the form \(\sum_{n=0}^\infty r^n\) where the common ratio \(r\) is \(\frac{(x+1)^{2}}{9}\).
Recall Theorem 20, which states that a geometric series \(\sum r^n\) converges if and only if \(|r| < 1\). Use this to find the interval of convergence by solving the inequality \(\left| \frac{(x+1)^2}{9} \right| < 1\).
Simplify the inequality to \(\frac{(x+1)^2}{9} < 1\), which leads to \((x+1)^2 < 9\). Then solve for \(x\) to find the interval where the series converges.
Within the interval of convergence, use the formula for the sum of a geometric series: \(S = \frac{a}{1 - r}\), where \(a\) is the first term of the series. Here, the first term corresponds to \(n=0\), so \(a = \frac{(x+1)^{0}}{9^{0}} = 1\).
Write the sum of the series as a function of \(x\) using the formula \(S(x) = \frac{1}{1 - \frac{(x+1)^2}{9}}\). Simplify this expression if desired to express the sum in a more compact form.
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Interval of Convergence
The interval of convergence is the set of x-values for which a power series converges. To find it, we typically use the Ratio or Root Test to determine where the series converges absolutely. This interval may be open, closed, or half-open depending on endpoint behavior.
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08:44
Interval of Convergence
Theorem 20 (Geometric Series Sum Formula)
Theorem 20 states that a geometric series ∑ arⁿ converges to a/(1 - r) if |r| < 1. Recognizing the given series as geometric allows us to find its sum function by identifying the common ratio and applying this formula within the interval of convergence.
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Percorso guidato
06:00Geometric Series
Power Series and Function Representation
A power series represents a function as an infinite sum of terms involving powers of (x - c). Understanding how to express a series as a function helps in analyzing its behavior and finding closed-form expressions, which is essential for interpreting the sum within the interval of convergence.
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07:32Representing Functions as Power Series
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 ∞) (tanh n) / n²
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Direct Comparison Test
In Exercises 1–8, use the Direct Comparison Test to determine if each series converges or diverges.
∑ (from n=1 to ∞) cos²n / n^(3/2)
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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 = 8^(1/n)
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Domanda del libro di testo
Convergence and Divergence
Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = (2n + 2)! / (2n − 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 / (n² + 2n + 1)]
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