Domanda del libro di testo
23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.
∑ (k = 1 to ∞) k^(–1/5)
66
views
Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 10, Problema 10.3.75
72–86. Evaluating series Evaluate each series or state that it diverges.
∑ (k = 0 to ∞) (1/4)ᵏ × 5^(3 – k)
Guida verificata passo dopo passo1
Identify the general term of the series: \( a_k = \left(\frac{1}{4}\right)^k \times 5^{3-k} \).
Rewrite the term to express it as a single base raised to the power \(k\): \( a_k = 5^3 \times \left(\frac{1}{4} \times \frac{1}{5}\right)^k = 125 \times \left(\frac{1}{20}\right)^k \).
Recognize that the series is a geometric series with first term \( a_0 = 125 \) and common ratio \( r = \frac{1}{20} \).
Check the convergence of the geometric series by verifying if \( |r| < 1 \). Since \( \frac{1}{20} < 1 \), the series converges.
Use the formula for the sum of an infinite geometric series: \( S = \frac{a_0}{1 - r} = \frac{125}{1 - \frac{1}{20}} \) to express the sum.
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:
3mConcetti 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^k, and converges if the absolute value of the ratio |r| < 1. The sum can be calculated using the formula S = a / (1 - r).
Video consigliato:
Percorso guidato
06:00
Geometric Series
Convergence and Divergence of Infinite Series
An infinite series converges if the sum approaches a finite limit as the number of terms increases indefinitely. If it does not approach a finite value, the series diverges. Determining convergence is essential before evaluating the sum of an infinite series.
Video consigliato:
Percorso guidato
06:52Convergence of an Infinite Series
Manipulating Series Terms
Simplifying the general term of a series often involves algebraic manipulation, such as factoring constants or rewriting terms with exponents. This helps identify the type of series and the common ratio, facilitating the application of convergence tests and sum formulas.
Video consigliato:
Percorso guidato
06:00Geometric Series
Pratica correlata
Domanda del libro di testo
Growth rates of sequences
Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.
aₙ = (6ⁿ + 3ⁿ) / (6ⁿ + n¹⁰⁰)
70
views
Domanda del libro di testo
40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 1 to ∞) (1 + 2 / k)ᵏ
31
views
Domanda del libro di testo
21–42. Geometric series Evaluate each geometric series or state that it diverges.
35.∑ (k = 0 to ∞) 3(–π)^(–k)
74
views
Domanda del libro di testo
45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 2 to ∞) (−1)ᵏ · k · (k² + 1) / (k³ − 1)
65
views
Domanda del libro di testo
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{2ⁿ⁺¹3⁻ⁿ}
77
views