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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.R.57
Capitolo 10, Problema 10.R.57

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞)5ᵏ / 2²ᵏ⁺¹

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Identify the given series: \( \sum_{k=1}^{\infty} \frac{5^k}{2^{2k+1}} \). This is an infinite series where each term is \( a_k = \frac{5^k}{2^{2k+1}} \).
Rewrite the general term to simplify the expression. Notice that \( 2^{2k+1} = 2^{2k} \cdot 2^1 = 2 \cdot (2^2)^k = 2 \cdot 4^k \). So, \( a_k = \frac{5^k}{2 \cdot 4^k} = \frac{1}{2} \cdot \left( \frac{5}{4} \right)^k \).
Recognize that the series is a geometric series with common ratio \( r = \frac{5}{4} \) and first term \( a_1 = \frac{1}{2} \cdot \left( \frac{5}{4} \right)^1 \).
Recall the convergence criterion for a geometric series: it converges if and only if \( |r| < 1 \). Here, check the absolute value of the ratio \( |r| = \frac{5}{4} \).
Since \( |r| > 1 \), conclude that the geometric series diverges by the geometric series test.

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Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. Determining whether such a series converges means checking if the sum approaches a finite limit as the number of terms grows indefinitely. Understanding convergence is essential to analyze the behavior of series like the one given.
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Convergence of an Infinite Series

Geometric Series

A geometric series has terms that multiply by a constant ratio each time. It converges if the absolute value of this ratio is less than one, and its sum can be found using a specific formula. Recognizing the given series as geometric simplifies the convergence test.
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Geometric Series

Convergence Tests

Convergence tests, such as the Ratio Test or Root Test, help determine if a series converges or diverges. These tests analyze the limit of term ratios or roots to conclude about convergence. Choosing an appropriate test is key to solving series problems efficiently.
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Choosing a Convergence Test
Pratica correlata
Domanda del libro di testo

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.

aₙ = ((3n² + 2n + 1) · sin(n)) / (4n³ + n) (Hint: Use the Squeeze Theorem.)

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Domanda del libro di testo

Geometric sums

Evaluate the geometric sums

∑ (from k = 0 to 9) (0.2)ᵏand∑ (from k = 2 to 9) (0.2)ᵏ.

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Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.

a.The terms of the sequence {aₙ} increase in magnitude, so the limit of the sequence does not exist.

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12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.

aₙ = 8ⁿ / n!

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42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.

∑ (from k = 1 to ∞)(7 + sin k) / k²

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Estimate the value of the series ∑ (from k = 1 to ∞)1 / (2k + 5)³ to within 10⁻⁴ of its exact value.

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