Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.5.21

Chapter 10, Problem 10.5.21

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.

∑ (k = 1 to ∞) ((3k³ + 4)(7k² + 1)) / ((2k³ + 1)(4k³ − 1))

Verified step by step guidance

First, identify the general term of the series: $a_k = \frac{(3k^{3} + 4)(7k^{2} + 1)}{(2k^{3} + 1)(4k^{3} - 1)}$.

Next, analyze the behavior of $a_k$ for large $k$ by focusing on the highest degree terms in numerator and denominator: numerator behaves like $3k^{3} \times 7k^{2} = 21k^{5}$, denominator behaves like $2k^{3} \times 4k^{3} = 8k^{6}$.

Simplify the dominant term ratio for large $k$: $a_k \sim \frac{21k^{5}}{8k^{6}} = \frac{21}{8k}$, which suggests $a_k$ behaves like $\frac{C}{k}$ for some constant $C$ as $k \to \infty$.

Choose a comparison series $b_k = \frac{1}{k}$, which is a $p$-series with $p=1$, known to diverge.

Apply the Limit Comparison Test by computing $\lim_{k \to \infty} \frac{a_k}{b_k} = \lim_{k \to \infty} a_k \times k$, and analyze the limit to determine if it is finite and positive, which will indicate whether $\sum a_k$ converges or diverges.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Comparison Test

The Comparison Test determines the convergence of a series by comparing it to a second series with known behavior. If the terms of the given series are smaller than those of a convergent series, it also converges. Conversely, if the terms are larger than those of a divergent series, it diverges. This test requires finding a suitable comparison series.

Limit Comparison Test

The Limit Comparison Test involves taking the limit of the ratio of the terms of two series. If this limit is a positive finite number, both series either converge or diverge together. This test is useful when direct comparison is difficult but the terms behave similarly for large indices.

Asymptotic Behavior of Rational Functions

Analyzing the dominant terms in the numerator and denominator of rational expressions helps simplify the general term of a series. For large k, lower-degree terms become negligible, allowing approximation of the term’s behavior. This simplification is crucial for choosing an appropriate comparison series.

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Related Practice

Textbook Question

32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (from k = 1 to ∞) k¹⁰⁰ / (k + 1)!

Textbook Question

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.

∑ (k = 1 to ∞) (2 + (−1)ᵏ) / k²

Textbook Question

16–17. {Use of Tech} Periodic savings

Suppose you deposit m dollars at the beginning of every month in a savings account that earns a monthly interest rate of r, which is the annual interest rate divided by 12 (for example, if the annual interest rate is 2.4%, r = 0.024/12 = 0.002). For an initial investment of m dollars, the amount of money in your account at the beginning of the second month is the sum of your second deposit and your initial deposit plus interest, or m + m(1 + r). Continuing in this fashion, it can be shown that the amount of money in your account after n months is

Aₙ = m + m(1 + r) + ⋯ + m(1 + r)ⁿ⁻¹.

Use geometric sums to determine the amount of money in your savings account after 5 years (60 months) using the given monthly deposit and interest rate.

17. Monthly deposits of \$250 at a monthly interest rate of 0.2%

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Textbook Question

13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.

aₙ = 1 + sin(πn / 2)

Textbook Question

Find a formula for the nth partial sum Sₙ of

∑ k = 1 to ∞[(1/(k + 3)) − (1/(k + 4))]

Use your formula to find the sum of the first 36 terms of the series.

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Textbook Question

72–86. Evaluating series Evaluate each series or state that it diverges.