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 + √n)]
26
views
Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
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 10.2.13
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)]
Guida verificata passo dopo passo1
Identify the general term of the series: \( a_n = \frac{1}{2^n} + \frac{(-1)^n}{5^n} \). This series is the sum of two separate geometric series.
Write out the first eight terms by substituting \( n = 0, 1, 2, \ldots, 7 \) into the general term: \( a_0, a_1, a_2, \ldots, a_7 \). This helps visualize how the series starts.
Recognize that each part is a geometric series: the first with common ratio \( r_1 = \frac{1}{2} \), and the second with common ratio \( r_2 = -\frac{1}{5} \). Both have initial terms \( a_0^{(1)} = 1 \) and \( a_0^{(2)} = 1 \) respectively.
Check the absolute values of the common ratios to determine convergence: since \( |r_1| < 1 \) and \( |r_2| < 1 \), both geometric series converge.
Use the formula for the sum of an infinite geometric series \( S = \frac{a}{1 - r} \) to find the sum of each series separately, then add these sums to get the total sum of the original series.
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
10mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Geometric Series
A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio. It has the form ∑ ar^n, where a is the first term and r is the common ratio. Understanding this helps in identifying and summing series with terms like (1/2)^n or (-1/5)^n.
Video consigliato:
Percorso guidato
06:00
Geometric Series
Sum of an Infinite Geometric Series
An infinite geometric series converges if the absolute value of the common ratio |r| is less than 1. Its sum is given by S = a / (1 - r). This formula is essential for finding the sum of each geometric component in the series when it converges.
Video consigliato:
Percorso guidato
06:00Geometric Series
Series Divergence and Convergence
A series converges if its partial sums approach a finite limit; otherwise, it diverges. For series composed of multiple parts, each part must be analyzed separately. Recognizing divergence or convergence is key to determining whether the sum exists or not.
Video consigliato:
Percorso guidato
06:52Convergence of an Infinite Series
Pratica correlata
Domanda del libro di testo
Finding Taylor Series at x = 0 (Maclaurin Series)
Find the Maclaurin series for the functions in Exercises 11–24.
5 cos πx
26
views
Domanda del libro di testo
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)
26
views
Domanda del libro di testo
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
42
views
Domanda del libro di testo
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
5
views
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 (ⁿ√10)]
26
views