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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 10.2.19
Chapter 10, Problem 10.2.19

In Exercises 15–22, determine if the geometric series converges or diverges. If a series converges, find its sum.
1 − (2/e) + (2/e)² − (2/e)³ + (2/e)⁴ − …

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Identify the first term \( a \) and the common ratio \( r \) of the geometric series. The series is \( 1 - \frac{2}{e} + \left(\frac{2}{e}\right)^2 - \left(\frac{2}{e}\right)^3 + \left(\frac{2}{e}\right)^4 - \cdots \), so \( a = 1 \) and \( r = -\frac{2}{e} \).
Recall that a geometric series \( \sum_{n=0}^\infty ar^n \) converges if and only if the absolute value of the common ratio is less than 1, i.e., \( |r| < 1 \).
Calculate \( |r| = \left| -\frac{2}{e} \right| = \frac{2}{e} \). Since \( e \approx 2.718 \), compare \( \frac{2}{e} \) to 1 to determine convergence.
If the series converges, use the formula for the sum of an infinite geometric series: \[ S = \frac{a}{1 - r} \]. Substitute the values of \( a \) and \( r \) into this formula.
Write the sum expression explicitly as \( S = \frac{1}{1 - \left(-\frac{2}{e}\right)} = \frac{1}{1 + \frac{2}{e}} \), which can be simplified further if needed.

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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 a sum of terms where each term is found by multiplying the previous term by a constant ratio, r. It has the form a + ar + ar² + ar³ + ..., where a is the first term. Understanding this structure is essential to analyze the given series.
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Convergence Criteria for Geometric Series

A geometric series converges if the absolute value of the common ratio |r| is less than 1. If |r| ≥ 1, the series diverges. This criterion helps determine whether the infinite sum approaches a finite value.
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Sum of a Convergent Geometric Series

If a geometric series converges, its sum can be calculated using the formula S = a / (1 - r), where a is the first term and r is the common ratio. This formula provides a direct way to find the total sum of the infinite series.
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Related Practice
Textbook Question

Repeating Decimals

Express each of the numbers in Exercises 23–30 as the ratio of two integers.

3.1̅4̅2̅8̅5̅7 = 3.142857142857 ...

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 ∞) eⁿ / (1 + e²ⁿ)

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Textbook Question

Finding Terms of a Sequence

Each of Exercises 1–6 gives a formula for the nth term aₙ of a sequence {aₙ}. Find the values of a₁, a₂, a₃, and a₄.

aₙ = 1 / n!

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Textbook Question

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

∑ (from n = 2 to ∞) [(ln n / n)³]

Textbook Question

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

∑ (from n = 1 to ∞) [3ⁿ / n³]

Textbook Question

Error Estimation

In Exercises 49–52, estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series.

1 / (1 + t) = ∑ (from n = 0 to ∞) [(-1)ⁿ tⁿ],0 < t < 1