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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblem 10.5.37
Chapter 10, Problem 10.5.37

Determining Convergence or Divergence
In Exercises 17–46, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑(from n=1 to ∞) [n! / (2n + 1)!]

Verified step by step guidance
1
First, identify the general term of the series: \(a_n = \frac{n!}{(2n + 1)!}\).
Since the terms involve factorials, consider using the Ratio Test, which is effective for series with factorial expressions.
Write the Ratio Test limit: \(L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \frac{(n+1)!}{(2(n+1) + 1)!} \cdot \frac{(2n + 1)!}{n!}\).
Simplify the factorial expressions inside the limit by expanding \((n+1)! = (n+1) \cdot n!\) and \((2(n+1) + 1)! = (2n + 3)! = (2n + 3)(2n + 2)(2n + 1)!\).
After simplification, evaluate the limit \(L\). If \(L < 1\), the series converges absolutely; if \(L > 1\), it diverges; if \(L = 1\), the test is inconclusive.

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Key Concepts

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

Factorial Growth and Comparison

Factorials grow very rapidly compared to polynomial or exponential functions. Understanding how n! compares to (2n+1)! is crucial for analyzing the behavior of the terms in the series and determining if they approach zero fast enough for convergence.
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Factorials

Ratio Test for Series Convergence

The Ratio Test involves taking the limit of the absolute value of the ratio of consecutive terms. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges. This test is especially useful for series involving factorials.
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Definition of Convergence and Divergence of Infinite Series

A series converges if the sequence of its partial sums approaches a finite limit; otherwise, it diverges. Recognizing this fundamental definition helps in applying tests and interpreting their results correctly.
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Convergence of an Infinite Series
Related Practice
Textbook Question

Determining Convergence or Divergence

In Exercises 17–46, use any method to determine whether the series converges or diverges. Give reasons for your answer.

∑(from n=3 to ∞) [2n² / n²ⁿ]

Textbook Question

Find the first four nonzero terms in the Maclaurin series for the functions in Exercises 31–38.

(tan⁻¹x)²

Textbook Question

In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.

∑ (from n = 1 to ∞) tan(n^(1/n))

Textbook Question

In Exercises 35–40, find the first three nonzero terms of the Maclaurin series for each function.

f(x) = (sin x) ln(1 + x)

Textbook Question

In Exercises 81–86, find the values of x for which the given geometric series converges.

Also, find the sum of the series (as a function of x) for those values of x.

∑ (from n = 0 to ∞) [ (−1/2)ⁿ (x − 3)ⁿ ]

Textbook Question

Use power series operations to find the Taylor series at x = 0 for the functions in Exercises 13–30.

sin x – x + (x³ / 3!)