Ch. 10 - Infinite Sequences and Series

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 10 - Infinite Sequences and Series

Problema 10.7.50

Capitolo 10, Problema 10.7.50

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

Guida verificata passo dopo passo

Recognize that the given series is a geometric series of the form \(\sum_{n=0}^\infty r^n\) where the common ratio \(r = \frac{(x^2 - 1)}{2}\).

Recall Theorem 20, which states that a geometric series \(\sum_{n=0}^\infty r^n\) converges if and only if \(|r| < 1\). Use this to find the interval of convergence by solving the inequality \(\left| \frac{(x^2 - 1)}{2} \right| < 1\).

Multiply both sides of the inequality by 2 to get \(|x^2 - 1| < 2\). This inequality will help determine the values of \(x\) for which the series converges.

Solve the inequality \(|x^2 - 1| < 2\) by considering the two cases: \(-2 < x^2 - 1 < 2\). Add 1 to all parts to get \(-1 < x^2 < 3\). Since \(x^2\) is always non-negative, the lower bound \(-1 < x^2\) is always true, so focus on \(x^2 < 3\) which implies \(-\sqrt{3} < x < \sqrt{3}\).

Within this interval, use the formula for the sum of a geometric series \(S = \frac{1}{1 - r}\), substituting \(r = \frac{(x^2 - 1)}{2}\) to express the sum as a function of \(x\).

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Interval of Convergence

The interval of convergence is the set of x-values for which a given power series converges. To find it, we often 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

Geometric Series and Its Sum

A geometric series has the form ∑ arⁿ and converges when |r| < 1. Its sum is given by a/(1 - r). Recognizing a series as geometric allows us to find a closed-form expression for the sum within the interval of convergence.

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Percorso guidato

06:00

Geometric Series

Theorem 20 (Power Series Convergence and Sum)

Theorem 20 typically states conditions under which a power series converges and how to express its sum as a function within the interval of convergence. Applying this theorem helps identify the radius and interval of convergence and find the sum function explicitly.

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Intro to Power Series

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Intervals of Convergence

In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?

∑ (from n = 1 to ∞) [ (ln n) xⁿ ]

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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=2 to ∞) [(3ⁿ⁺²) / ln(n)]

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Make up a geometric series ∑a rⁿ⁻¹ that converges to the number 5 if

b. a = 13/2

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