Skip to main content
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 10.3.23
Chapter 10, Problem 10.3.23

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


23.∑ (k = 0 to ∞) (–9/10)ᵏ

Verified step by step guidance
1
Identify the first term \( a \) and the common ratio \( r \) of the geometric series. Here, the series is \( \sum_{k=0}^{\infty} \left(-\frac{9}{10}\right)^k \), so \( a = 1 \) (when \( k=0 \)) and \( r = -\frac{9}{10} \).
Recall the formula for the sum of an infinite geometric series: if \( |r| < 1 \), then the series converges and its sum is \( S = \frac{a}{1 - r} \). Otherwise, the series diverges.
Check the absolute value of the common ratio: \( |r| = \left| -\frac{9}{10} \right| = \frac{9}{10} < 1 \). Since this is true, the series converges.
Apply the sum formula by substituting \( a = 1 \) and \( r = -\frac{9}{10} \) into \( S = \frac{a}{1 - r} \). This gives \( S = \frac{1}{1 - \left(-\frac{9}{10}\right)} \).
Simplify the denominator to find the expression for the sum: \( 1 - \left(-\frac{9}{10}\right) = 1 + \frac{9}{10} \). This completes the setup to calculate the sum.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

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. It is expressed as ∑ ar^k, where a is the first term and r is the common ratio.
Recommended video:
06:00
Geometric Series

Convergence and Divergence of Infinite Series

An infinite geometric series converges if the absolute value of the common ratio |r| is less than 1, meaning the sum approaches a finite limit. If |r| ≥ 1, the series diverges and does not have a finite sum.
Recommended video:
06:52
Convergence of an Infinite Series

Sum Formula for Convergent Geometric Series

For a convergent geometric series with first term a and common ratio r (|r| < 1), the sum to infinity is given by S = a / (1 - r). This formula allows direct calculation of the series sum without adding infinitely many terms.
Recommended video:
06:00
Geometric Series
Related Practice
Textbook Question

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


41.∑ (k = 1 to ∞) 4 / 12ᵏ

Textbook Question

11–27. Alternating Series Test Determine whether the following series converge.

∑ (k = 1 to ∞) (−1)ᵏ⁺¹ k² / (k³ + 1)

Textbook Question

33–38. {Use of Tech} Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10⁻⁴ in magnitude. Although you do not need it, the exact value of the series is given in each case.


ln 2 = ∑ (k = 1 to ∞) (−1)ᵏ⁺¹ / k

1
views
Textbook Question

55–70. More sequences

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


{cosn / n}

1
views
Textbook Question

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

∑ (from k = 1 to ∞)sin(1 / k⁹)

Textbook Question

46–53. Decimal expansions

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers).


51.0.456̅ = 0.456456456…