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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.3.39
Capitolo 10, Problema 10.3.39

21–42. Geometric series Evaluate each geometric series or state that it diverges.  


39.∑ (k = 2 to ∞) (–0.15)ᵏ

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Identify the first term of the geometric series by substituting the lower limit of the summation into the general term: the first term is \(a = (-0.15)^2\).
Determine the common ratio \(r\) of the geometric series, which is the base of the exponent in the term, so \(r = -0.15\).
Check the convergence of the series by evaluating the absolute value of the common ratio: if \(|r| < 1\), the series converges; otherwise, it diverges.
Since the series starts at \(k=2\) instead of \(k=0\) or \(k=1\), express the sum from \(k=2\) to infinity in terms of the sum from \(k=0\) to infinity by factoring out the first two terms or adjusting the formula accordingly.
Use the formula for the sum of an infinite geometric series \(S = \frac{a}{1 - r}\), where \(a\) is the first term of the series starting at \(k=2\), to write the sum expression without calculating the final numerical value.

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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. Understanding the structure helps determine convergence and sum.
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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. If |r| ≥ 1, the series diverges. This criterion is essential to decide whether the series has a finite sum or not.
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Convergence of an Infinite Series

Sum Formula for Convergent Geometric Series

When an infinite geometric series converges, its sum is given by S = a / (1 - r), where a is the first term and r is the common ratio. This formula allows direct calculation of the series sum once convergence is established.
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Geometric Series
Pratica correlata
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9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.

∑ (k = 2 to ∞) √k / (ln¹⁰ k)

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11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.


∑ (from k = 1 to ∞)1 / ln(eᵏ + 1)

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Is it possible for a series of positive terms to converge conditionally? Explain.

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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.

∑ (from k = 1 to ∞) (2ᵏ / k⁹⁹)

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35–44. Limits of sequences Write the terms a₁, a₂, a₃, and a₄ of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. 

aₙ = 3 + cos(π*ⁿ) ; n = 1, 2, 3, …

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Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.


{nsin(6 / n)}

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