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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblema 10.7.28c
Capitolo 10, Problema 10.7.28c

Intervals of Convergence
In Exercises 1–36, for what values of x does the series converge (c) conditionally?
∑ (from n = 0 to ∞) [ (−2)ⁿ (n + 1) (x − 1)ⁿ ]

Guida verificata passo dopo passo
1
Identify the given power series: \(\sum_{n=0}^{\infty} (-2)^n (n+1) (x-1)^n\).
Rewrite the series in a form that isolates the variable term: \(\sum_{n=0}^{\infty} (n+1) \left[-2(x-1)\right]^n\).
Use the Root Test or Ratio Test to find the radius of convergence. For the Ratio Test, consider the limit \(L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\), where \(a_n = (n+1) \left[-2(x-1)\right]^n\).
Solve the inequality \(L < 1\) to find the interval of convergence in terms of \(x\).
Check the endpoints of the interval separately by substituting them back into the original series to determine if the series converges absolutely, conditionally, or diverges at those points.

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Radius and Interval of Convergence

The radius of convergence determines the distance from the center point within which a power series converges absolutely. The interval of convergence includes all x-values for which the series converges, possibly including endpoints where convergence must be tested separately.
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07:36
Radius of Convergence

Absolute vs. Conditional Convergence

A series converges absolutely if the series of absolute values converges; otherwise, it may converge conditionally if the original series converges but not absolutely. Conditional convergence often occurs at the endpoints of the interval of convergence.
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Percorso guidato
07:51
Choosing a Convergence Test

Ratio Test for Convergence

The ratio test uses the limit of the absolute value of the ratio of consecutive terms to determine convergence. If this limit is less than one, the series converges absolutely; if greater than one, it diverges; if equal to one, the test is inconclusive and further analysis is needed.
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Percorso guidato
05:35
Ratio Test
Pratica correlata
Domanda del libro di testo

Assume that the series ∑ aₙ(x − 2)ⁿ converges for x = −1 and diverges for x = 6. Answer true (T), false (F), or not enough information given (N) for the following statements about the series.

f. Diverges for x = 4.9

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Domanda del libro di testo

Intervals of Convergence

In Exercises 1–36, for what values of x does the series converge (c) conditionally?

∑ (from n = 1 to ∞) [ (3x + 1)^(n + 1) / (2n + 2) ]

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Domanda del libro di testo

A sequence of rational numbers is described as follows:

1/1,3/2,7/5,17/12,…,a/b,(a + 2b)/(a + b),…

Here the numerators form one sequence, the denominators form a second sequence, and their ratios form a third sequence. Let xₙ and yₙ be, respectively, the numerator and the denominator of the nᵗʰ fraction rₙ = xₙ / yₙ.

b. The fractions rₙ = xₙ / yₙ approach a limit as n increases. What is that limit? (Hint: Use part (a) to show that rₙ² − 2 = ±(1 / yₙ)² and that yₙ is not less than n.)

33
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Domanda del libro di testo

Assume that the series ∑ aₙxⁿ converges for x = 4 and diverges for x = 7. Answer true (T), false (F), or not enough information given (N) for the following statements about the series.

e. Diverges for x = 8

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Domanda del libro di testo

Intervals of Convergence

In Exercises 1–36, for what values of x does the series converge (c) conditionally?

∑ (from n = 1 to ∞) [ (√(n + 1) − √n)(x − 3)ⁿ ]

22
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