Textbook Question
Estimate the value of ∑ (from n=2 to ∞) (1 / (n² + 4)) to within 0.1 of its exact value.
Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
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 − …
Verified step by step guidance1
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
Finding Taylor Polynomials
In Exercises 1–10, find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a.
f(x) = √(1 − x),a = 0
Textbook Question
Determining Convergence or Divergence
In Exercises 1–14, determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.
∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (1 / n^(3/2))]
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Textbook Question
Find the sum of each series in Exercises 45–52.
∑ (from n = 1 to ∞) [ (2n + 1) / (n²(n + 1)²) ]
Textbook Question
In Exercises 121–124, determine whether the sequence is monotonic and whether it is bounded.
aₙ = 2ⁿ 3ⁿ / n!
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Textbook Question
Finding Taylor and Maclaurin Series
In Exercises 25–34, find the Taylor series generated by f at x = a.
f(x) = cos(2x + π/2),a = π/4