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 ∞) (2k + 1)! / (k!)²

검증된 단계별 안내

First, identify the general term of the series: \(a_k = \frac{(2k + 1)!}{(k!)^2}\).

To determine convergence, consider using the Ratio Test, which is effective for factorial expressions. The Ratio Test states to compute \(L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|\).

Write out the ratio \(\frac{a_{k+1}}{a_k} = \frac{(2(k+1) + 1)!}{((k+1)!)^2} \times \frac{(k!)^2}{(2k + 1)!}\) and simplify the factorial expressions carefully.

Evaluate the limit \(L\) as \(k\) approaches infinity. If \(L < 1\), the series converges absolutely; if \(L > 1\), the series diverges; if \(L = 1\), the test is inconclusive.

Based on the result of the Ratio Test, conclude whether the series converges absolutely, converges conditionally, or diverges.

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주요 개념

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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.

Ratio Test

The Ratio Test determines convergence by examining the limit of the ratio of consecutive terms. If the limit is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive.

Factorials and Growth Rates

Factorials grow very rapidly, often dominating polynomial terms. Comparing factorial expressions using simplification or the Ratio Test helps analyze the behavior of series terms and determine convergence or divergence.

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 ∞) (2k + 1)! / (k!)²