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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.3.9
Chapter 10, Problem 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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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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
Related Practice
Textbook Question

Suppose the sequence { aₙ} is defined by the explicit formula aₙ = 1/n, for n=1, 2, 3, .....Write out the first five terms of the sequence.

Textbook Question

54–69. Telescoping series

For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.


57. ∑ (k = 1 to ∞) 1 / ((k + 6)(k + 7))

Textbook Question

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

∑ (from k = 1 to ∞) (10ᵏ + 1) / k¹⁰

Textbook Question

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 ∞) (−1)ᵏ k (2ᵏ⁺¹ / (9ᵏ − 1))

Textbook Question

1–10. Choosing convergence tests Identify a convergence test for each series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.

∑ (from k = 1 to ∞) ((−1)ᵏ⁺¹) / (√2ᵏ + lnk)

Textbook Question

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

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