11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) 5ᵏ(k!)² / (2k)!

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Identify the series given: \( \sum_{k=1}^{\infty} \frac{5^{k} (k!)^{2}}{(2k)!} \). We want to determine if this infinite series converges or diverges.

Consider using the Ratio Test, which is often effective for series involving factorials and exponentials. The Ratio Test states that for \( a_k \), if \( L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \), then the series converges if \( L < 1 \), diverges if \( L > 1 \), and is inconclusive if \( L = 1 \).

Write the general term \( a_k = \frac{5^{k} (k!)^{2}}{(2k)!} \) and compute the ratio \( \frac{a_{k+1}}{a_k} = \frac{5^{k+1} ((k+1)!)^{2} / (2(k+1))!}{5^{k} (k!)^{2} / (2k)!} \). Simplify this expression step-by-step.

Simplify the ratio by canceling common terms and expressing factorials explicitly: \( \frac{a_{k+1}}{a_k} = 5 \times \frac{((k+1)!)^{2}}{(k!)^{2}} \times \frac{(2k)!}{(2k+2)!} \). Use the factorial properties \( (k+1)! = (k+1)k! \) and \( (2k+2)! = (2k+2)(2k+1)(2k)! \) to rewrite the ratio.

After simplification, take the limit as \( k \to \infty \) of the ratio. Analyze the behavior of the resulting expression to determine if the limit is less than, equal to, or greater than 1. This will tell you whether the series converges or diverges by the Ratio Test.

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Ratio Test for Convergence

The Ratio Test determines the convergence of a series by examining 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; if equal to 1, the test is inconclusive. This test is especially useful for series involving factorials and exponential terms.

Factorials and Their Growth Rates

Factorials (n!) represent the product of all positive integers up to n and grow very rapidly. Understanding how factorials compare to exponential functions is crucial when analyzing series terms, as factorial growth often dominates polynomial or exponential growth, affecting convergence behavior.

Absolute Convergence of Series

A series converges absolutely if the series of absolute values of its terms converges. Absolute convergence guarantees convergence regardless of term signs and allows the use of tests like the Ratio Test. This concept is important when dealing with series that have factorials and powers, ensuring a robust conclusion about convergence.

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.∑ (from k = 1 to ∞) 5ᵏ(k!)² / (2k)!