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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblem 10.7.46
Chapter 10, Problem 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

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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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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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Geometric 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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Representing Functions as Power Series
Related Practice
Textbook Question

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 ∞) [ (x² − 1) / 2 ]ⁿ

Textbook Question

If ∑aₙ is a convergent series of positive terms, prove that ∑sin(aₙ) converges.

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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 ∞) (tanh n) / n²

Textbook Question

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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Textbook Question

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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Textbook Question

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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