Domanda del libro di testo
∑ (from n=1 to ∞) (1 / √(n + 1)) diverges
b. What should n be in order that the partial sum sₙ = ∑ (from i=1 to n) (1 / √(i + 1)) satisfies sₙ > 1000?
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Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 10, Problema 10.4.74b
b. From Example 5, Section 10.2, show that
S = 1 + ∑(from n=1 to ∞) [1 / (n²(n + 1))].
Guida verificata passo dopo passo1
Recall the series expression from Example 5, Section 10.2, which involves the sum of terms related to \( \frac{1}{n^2 (n+1)} \). Our goal is to express \( S \) as \( 1 + \sum_{n=1}^\infty \frac{1}{n^2 (n+1)} \).
Start by writing the general term of the series: \( \frac{1}{n^2 (n+1)} \). To simplify or analyze this term, consider using partial fraction decomposition to break it into simpler fractions that are easier to sum.
Set up the partial fraction decomposition for \( \frac{1}{n^2 (n+1)} \) as \( \frac{A}{n} + \frac{B}{n^2} + \frac{C}{n+1} \), and solve for constants \( A, B, C \) by multiplying both sides by \( n^2 (n+1) \) and equating coefficients.
Once the partial fractions are found, rewrite the sum \( \sum_{n=1}^\infty \frac{1}{n^2 (n+1)} \) as the sum of simpler series involving \( \sum \frac{1}{n} \), \( \sum \frac{1}{n^2} \), and \( \sum \frac{1}{n+1} \).
Recognize that the term \( 1 \) outside the summation corresponds to the initial term or a boundary condition from the original series, completing the expression \( S = 1 + \sum_{n=1}^\infty \frac{1}{n^2 (n+1)} \).
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Infinite Series and Summation Notation
An infinite series is the sum of infinitely many terms, often expressed using summation notation (∑). Understanding how to interpret and manipulate these sums is essential for evaluating or transforming series expressions.
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Partial fraction decomposition breaks a complex rational expression into simpler fractions. This technique is useful for simplifying terms like 1/(n²(n+1)) to facilitate summation or integration.
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Telescoping Series
A telescoping series is one where many terms cancel out when the series is expanded, leaving only a few terms to sum. Recognizing and applying telescoping can simplify the evaluation of series like the one given.
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Pratica correlata
Domanda del libro di testo
A sequence of rational numbers is described as follows:
1/1,3/2,7/5,17/12,…,a/b,(a + 2b)/(a + b),…
Here the numerators form one sequence, the denominators form a second sequence, and their ratios form a third sequence. Let xₙ and yₙ be, respectively, the numerator and the denominator of the nᵗʰ fraction rₙ = xₙ / yₙ.
b. The fractions rₙ = xₙ / yₙ approach a limit as n increases. What is that limit? (Hint: Use part (a) to show that rₙ² − 2 = ±(1 / yₙ)² and that yₙ is not less than n.)
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Domanda del libro di testo
Intervals of Convergence
In Exercises 1–36, for what values of x does the series converge (b) absolutely?
∑ (from n = 1 to ∞) [ (√(n + 1) − √n)(x − 3)ⁿ ]
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Domanda del libro di testo
Use the Cauchy condensation test from Exercise 59 to show that:
b. ∑ (from n=1 to ∞) [1 / nᵖ] converges if p > 1 and diverges if p ≤ 1.
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Domanda del libro di testo
Intervals of Convergence
Intervals of Convergence
In Exercises 1–36, for what values of x does the series converge (b) absolutely?
∑ (from n = 1 to ∞) [ (3x + 1)^(n + 1) / (2n + 2) ]
29
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Domanda del libro di testo
Intervals of Convergence
In Exercises 1–36, for what values of x does the series converge (b) absolutely?
∑ (from n = 0 to ∞) [ (−2)ⁿ (n + 1) (x − 1)ⁿ ]
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