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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 10.3.13
Chapter 10, Problem 10.3.13

9–15. Geometric sums Evaluate each geometric sum.


{Use of Tech}∑ k = 0 to 9(−3/4)ᵏ

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Identify the type of series given. This is a geometric series where each term is of the form \(a r^k\), with \(a\) being the first term and \(r\) the common ratio.
Determine the first term \(a\) by substituting \(k=0\) into the term \(\left(-\frac{3}{4}\right)^k\). Since any number to the zero power is 1, \(a = 1\).
Identify the common ratio \(r\) as \(-\frac{3}{4}\), which is the base being raised to the power \(k\).
Use the formula for the sum of the first \(n+1\) terms of a geometric series: \(S_{n} = a \frac{1 - r^{n+1}}{1 - r}\) where \(n=9\) in this problem.
Substitute \(a = 1\), \(r = -\frac{3}{4}\), and \(n = 9\) into the formula to express the sum as \(S_9 = \frac{1 - \left(-\frac{3}{4}\right)^{10}}{1 - \left(-\frac{3}{4}\right)}\).

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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. The series has the form ∑ ar^k, where a is the first term and r is the common ratio.
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Geometric Series

Sum Formula for Finite Geometric Series

The sum of the first n+1 terms of a geometric series is given by S_n = a(1 - r^(n+1)) / (1 - r), provided r ≠ 1. This formula allows quick calculation of the sum without adding each term individually.
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Use of Technology in Calculations

Technology such as calculators or software can efficiently compute sums of series, especially when dealing with complex ratios or many terms. It helps verify manual calculations and handle more complicated expressions.
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Using The Velocity Function
Related Practice
Textbook Question

Is it possible for an alternating series to converge absolutely but not conditionally?

Textbook Question

Periodic doses

Suppose you take 200 mg of an antibiotic every 6 hr. The half-life of the drug (the time it takes for half of the drug to be eliminated from your blood) is 6 hr. Use infinite series to find the long-term (steady-state) amount of antibiotic in your blood. You may assume the steady-state amount is finite.

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Textbook Question

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

∑ (k = 2 to ∞) (5lnk) / k

Textbook Question

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

∑ (k = 1 to ∞) tan(1 / k)

Textbook Question

61–66. Sequences of partial sums For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series or state that the series diverges.


4 + 0.9 + 0.09 + 0.009 + ⋯  

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Textbook Question

45–48. {Use of Tech} Explicit formulas for sequences Consider the formulas for the following sequences {aₙ}ₙ₌₁∞

 Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.

aₙ = ⁿ² + n ;n = 1, 2, 3, …

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