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 10.3.7
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.
Verified step by step guidance1
Recognize that the series is a telescoping series of the form \( \sum_{k=1}^\infty \left( \frac{1}{k+3} - \frac{1}{k+4} \right) \). This means many terms will cancel out when we write out the partial sums explicitly.
Write the nth partial sum \( S_n \) as \( S_n = \sum_{k=1}^n \left( \frac{1}{k+3} - \frac{1}{k+4} \right) \).
Expand the sum to see the cancellation pattern: \( S_n = \left( \frac{1}{4} - \frac{1}{5} \right) + \left( \frac{1}{5} - \frac{1}{6} \right) + \cdots + \left( \frac{1}{n+3} - \frac{1}{n+4} \right) \). Notice that most intermediate terms cancel out.
After cancellation, the partial sum simplifies to \( S_n = \frac{1}{4} - \frac{1}{n+4} \). This is the formula for the nth partial sum.
To find the sum of the first 36 terms, substitute \( n = 36 \) into the formula: \( S_{36} = \frac{1}{4} - \frac{1}{36+4} \). You can then simplify this expression to get the numerical value.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Telescoping Series
A telescoping series is a series where many terms cancel out when the partial sums are expanded, leaving only a few terms from the beginning and end. This simplification makes it easier to find a formula for the nth partial sum by identifying the pattern of cancellation.
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Geometric Series
Partial Sums
The nth partial sum, Sₙ, is the sum of the first n terms of a series. Finding a formula for Sₙ helps analyze the behavior of the series and compute sums for specific values of n without adding each term individually.
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08:01Integration Using Partial Fractions
Evaluating Finite Sums
Once a formula for the nth partial sum is found, it can be used to calculate the sum of a finite number of terms by substituting the desired n value. This process avoids direct summation and leverages the simplified expression derived from the telescoping property.
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Related Practice
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
72–86. Evaluating series Evaluate each series or state that it diverges.
∑ (k = 1 to ∞) (((1/6)ᵏ + (1/3)ᵏ) × k⁻¹)
Textbook Question
40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 1 to ∞) k⁸ / (k¹¹ + 3)
Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
1 / (2·3) + 1 / (4·5) + 1 / (6·7) + 1 / (8·9) + ⋯