Domanda del libro di testo
Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{nsin(6 / n)}
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Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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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 10.3.25
21–42. Geometric series Evaluate each geometric series or state that it diverges.
25.∑ (k = 0 to ∞) 0.9ᵏ
Guida verificata passo dopo passo1
Identify the type of series given. The series is a geometric series of the form \(\sum_{k=0}^{\infty} ar^k\), where \(a\) is the first term and \(r\) is the common ratio.
Determine the first term \(a\) and the common ratio \(r\). Here, \(a = 0.9^0 = 1\) and \(r = 0.9\).
Check the convergence condition for a geometric series. A geometric series converges if and only if \(|r| < 1\).
Since \(|0.9| < 1\), the series converges. Use the formula for the sum of an infinite geometric series: \(S = \frac{a}{1 - r}\).
Substitute the values of \(a\) and \(r\) into the formula to express the sum: \(S = \frac{1}{1 - 0.9}\). This expression represents the sum of the series.
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Geometric Series
A geometric series is the sum of the terms of a geometric sequence, where each term is found by multiplying the previous term by a constant ratio. It is expressed as ∑ ar^k, where a is the first term and r is the common ratio.
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Percorso guidato
06:00
Geometric Series
Convergence of Infinite Geometric Series
An infinite geometric series converges if the absolute value of the common ratio |r| is less than 1. When it converges, the sum can be calculated using the formula S = a / (1 - r). If |r| ≥ 1, the series diverges.
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Percorso guidato
06:52Convergence of an Infinite Series
Sum Formula for Infinite Geometric Series
For a convergent infinite geometric series with first term a and common ratio r (|r| < 1), the sum is given by S = a / (1 - r). This formula allows quick evaluation of the series without summing infinitely many terms.
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Percorso guidato
06:00Geometric Series
Pratica correlata
Domanda del libro di testo
Does a geometric series always have a finite value?
100
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Domanda del libro di testo
11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 1 to ∞) (−1)ᵏ (k¹¹ + 2k⁵ + 1) / [4k(k¹⁰ + 1)]
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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 ∞) 1 / ∛k
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Domanda del libro di testo
55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.
{nsin³(nπ / 2) / (n + 1)}"
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Domanda del libro di testo
9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.
1 + (1 / 2)² + (1 / 3)³ + (1 / 4)⁴ + ⋯
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