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)!]

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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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주요 개념

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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.

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.

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.

Determining Convergence or DivergenceIn 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)!]