Skip to main content
Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblem 10.2.59
Chapter 10, Problem 10.2.59

Which series in Exercises 53–76 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.
∑ (from n = 0 to ∞) e^(−2n)

Verified step by step guidance
1
Identify the type of series given: \( \sum_{n=0}^{\infty} e^{-2n} \). Notice that this is a geometric series because each term can be written as \( (e^{-2})^n \).
Recall the formula for the sum of an infinite geometric series: if \( |r| < 1 \), then \( \sum_{n=0}^{\infty} ar^n = \frac{a}{1-r} \), where \( a \) is the first term and \( r \) is the common ratio.
Determine the first term \( a \) and the common ratio \( r \) for this series. Here, \( a = e^{-2 \cdot 0} = 1 \) and \( r = e^{-2} \).
Check the convergence condition by verifying if \( |r| < 1 \). Since \( e^{-2} \) is a positive number less than 1, the series converges.
Use the sum formula to express the sum of the series as \( \frac{1}{1 - e^{-2}} \). This represents the sum of the infinite series.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
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 a series where each term is obtained by multiplying the previous term by a constant ratio r. It converges if the absolute value of r is less than 1, and its sum can be found using the formula S = a / (1 - r), where a is the first term.
Recommended video:
06:00
Geometric Series

Convergence Criteria for Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. For geometric series, this depends on the common ratio, while for other series, tests like the comparison or ratio test may be used to determine convergence or divergence.
Recommended video:
06:52
Convergence of an Infinite Series

Exponential Functions in Series

Exponential functions of the form e^(−kn) decrease rapidly as n increases when k > 0. When used as terms in a series, they often form geometric series with ratio e^(−k), which helps in analyzing convergence and calculating sums.
Recommended video:
6:13
Exponential Functions
Related Practice
Textbook Question

Determining Convergence or Divergence

Which of the series in Exercises 13–46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

∑ (from n=1 to ∞) (1 + 1/n)ⁿ

Textbook Question

Absolute and Conditional Convergence

Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.

∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (ⁿ√10)]

1
views
Textbook Question

Convergence and Divergence

Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.

aₙ = (1/n) ∫₁ⁿ (1/x) dx

1
views
Textbook Question

Does the series

∑ (from n=1 to ∞) (1/n − 1/n²)

converge or diverge? Justify your answer.

1
views
Textbook Question

In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.

∑ (from n = 0 to ∞) [((n + 1) / (n + 2))ⁿ]

Textbook Question

Intervals of Convergence

In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?

∑ (from n = 1 to ∞) [ (x − 1)ⁿ / √n ]