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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 10.3.9
Capitolo 10, Problema 10.3.9

9–15. Geometric sums Evaluate each geometric sum.


∑ k = 0 to 83ᵏ

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1
Identify the type of series given. Here, the sum is a geometric series because each term is obtained by multiplying the previous term by a constant ratio. The series is \( \sum_{k=0}^{8} 3^k \).
Recall the formula for the sum of the first \( n+1 \) terms of a geometric series: \[ S_{n} = \frac{a(r^{n+1} - 1)}{r - 1} \] where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the highest exponent index.
Identify the values for this problem: the first term \( a = 3^0 = 1 \), the common ratio \( r = 3 \), and the number of terms is \( n+1 = 9 \) since \( k \) goes from 0 to 8.
Substitute these values into the geometric sum formula: \[ S_8 = \frac{1 \times (3^{9} - 1)}{3 - 1} \]. This expression represents the sum of the series.
Simplify the denominator and prepare to evaluate the numerator (without calculating the final number), which completes the setup for finding the sum of the geometric 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. For example, in the series 3^0 + 3^1 + ... + 3^8, the common ratio is 3.
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Geometric Series

Formula for the Sum of a Finite Geometric Series

The sum of the first n+1 terms of a geometric series with initial term a and common ratio r (r ≠ 1) is given by S = a(r^(n+1) - 1) / (r - 1). This formula allows quick calculation without adding each term individually.
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Geometric Series

Exponentiation and Index Notation

Exponentiation involves raising a base to a power, indicating repeated multiplication. Understanding how to interpret and manipulate expressions like 3^k is essential for evaluating terms in the geometric sum.
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Sigma Notation
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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 ∞) ((-1)ᵏ⁺¹) × ((10k³ + k) / (9k³ + k + 1))ᵏ

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The terms of a sequence of partial sums are defined by Sₙ = ∑ⁿₖ₌₁ k² , for n=1, 2, 3, .....Evaluate the first four terms of the sequence.

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Domanda del libro di testo

40–62. Choose your test Use the test of your choice to determine whether the following series converge.

∑ (k = 2 to ∞) 1 / (klnk)

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Domanda del libro di testo

13–52. Limits of sequences

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


{(3n³ − 1)⁄(2n³ + 1)}

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Domanda del libro di testo

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

∑ (from k = 1 to ∞) (−1)ᵏ × k / (k³ + 1)

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Domanda del libro di testo

32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (from k = 1 to ∞) (√k / k − 1)²ᵏ

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