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 ∞) ((3k³ + 4)(7k² + 1)) / ((2k³ + 1)(4k³ − 1))
Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 10, Problem 10.7.37
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)!
Verified step by step guidance1
Identify the given series: \( \sum_{k=1}^{\infty} \frac{k^{100}}{(k+1)!} \). We want to determine if it converges absolutely, conditionally, or diverges.
Since the terms involve factorials, consider using the Ratio Test, which is effective for series with factorial expressions.
Set up the Ratio Test by examining the limit \( L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \), where \( a_k = \frac{k^{100}}{(k+1)!} \). Write \( a_{k+1} = \frac{(k+1)^{100}}{(k+2)!} \).
Compute the ratio \( \frac{a_{k+1}}{a_k} = \frac{(k+1)^{100}}{(k+2)!} \times \frac{(k+1)!}{k^{100}} \). Simplify the factorial terms and the powers to express the ratio in a form suitable for taking the limit as \( k \to \infty \).
Evaluate the limit \( L \). If \( L < 1 \), the series converges absolutely; if \( L > 1 \), it diverges; if \( L = 1 \), the test is inconclusive and another test must be used.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Absolute and Conditional Convergence
A series converges absolutely if the series of absolute values converges. If the original series converges but not absolutely, it converges conditionally. Understanding these distinctions helps classify the behavior of infinite series.
Recommended video:
07:51
Choosing a Convergence Test
Ratio Test
The Ratio Test determines convergence by examining the limit of the ratio of consecutive terms. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges. It is especially useful for series involving factorials.
Recommended video:
05:35Ratio Test
Factorials and Growth Rates
Factorials grow faster than any polynomial function. Recognizing how factorial terms dominate polynomial terms in the numerator or denominator helps predict the behavior of series and guides the choice of convergence tests.
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5:22Factorials
Related Practice
Textbook Question
11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 1 to ∞) (−1)ᵏ⁺¹ k¹/ᵏ
Textbook Question
21–42. Geometric series Evaluate each geometric series or state that it diverges.
29.∑ (k = 1 to ∞) e^(–2k)
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Textbook Question
9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.
∑ (k = 0 to ∞) 1 / (1000 + k)
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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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