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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.4.85
Chapter 10, Problem 10.4.85

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) + ⋯

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Identify the general term of the series. Notice the pattern in the denominators: the first term is \( \frac{1}{2 \cdot 3} \), the second is \( \frac{1}{4 \cdot 5} \), the third is \( \frac{1}{6 \cdot 7} \), and so on. We can express the \(n\)-th term as \( a_n = \frac{1}{(2n)(2n+1)} \).
Simplify the general term using partial fraction decomposition. Write \( \frac{1}{(2n)(2n+1)} \) as \( \frac{A}{2n} + \frac{B}{2n+1} \) and solve for constants \(A\) and \(B\). This will help in understanding the behavior of the series.
After finding the partial fractions, rewrite the series as a telescoping series if possible. Telescoping series have terms that cancel out sequentially, making it easier to analyze convergence.
Evaluate the partial sums of the series using the telescoping form. Write the sum of the first \(N\) terms and simplify to see if the sum approaches a finite limit as \(N \to \infty\).
Conclude about the convergence of the series based on the behavior of the partial sums. If the partial sums approach a finite number, the series converges; otherwise, it diverges.

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Key Concepts

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

Convergence of Infinite Series

An infinite series converges if the sum of its terms approaches a finite limit as the number of terms increases indefinitely. Understanding convergence is essential to determine whether adding infinitely many terms results in a finite value or diverges to infinity.
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Convergence of an Infinite Series

Comparison Test

The Comparison Test involves comparing the given series to a known benchmark series with established convergence behavior. If the terms of the given series are smaller than those of a convergent series, it also converges; if larger than a divergent series, it diverges.
Recommended video:
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Direct Comparison Test

General Term Analysis and Simplification

Expressing the general term of the series in a simplified form helps identify its behavior and apply convergence tests effectively. For example, rewriting terms like 1/(2n(2n+1)) can reveal similarities to p-series or telescoping series.
Recommended video:
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Divergence Test (nth Term Test)
Related Practice
Textbook Question

What comparison series would you use with the Comparison Test to determine whether ∑ (k = 1 to ∞) 2ᵏ / (3ᵏ + 1) converges?

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.

∑ (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

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

aₙ = (−1)ⁿ / 2ⁿ

Textbook Question

21–42. Geometric series Evaluate each geometric series or state that it diverges.  


37.1 + e/π + e²/π² + e³/π³ + ⋯

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