In Exercises 37–42, find the series’ radius of convergence.
∑ (from n = 1 to ∞) [ n! xⁿ / nⁿ ]

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Identify the general term of the series as \(a_n = \frac{n! x^n}{n^n}\).

Recall that the radius of convergence \(R\) can be found using the Root Test or the Ratio Test. Here, the Ratio Test is convenient, which involves evaluating \(\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\).

Write the ratio \(\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{(n+1)! x^{n+1} / (n+1)^{n+1}}{n! x^n / n^n} \right| = |x| \cdot \frac{(n+1)!}{n!} \cdot \frac{n^n}{(n+1)^{n+1}}\).

Simplify the factorial and powers: \(\frac{(n+1)!}{n!} = n+1\), so the ratio becomes \(|x| (n+1) \frac{n^n}{(n+1)^{n+1}}\).

Rewrite \(\frac{n^n}{(n+1)^{n+1}}\) as \(\frac{n^n}{(n+1)^n (n+1)} = \frac{1}{n+1} \left( \frac{n}{n+1} \right)^n\), then simplify the entire expression and take the limit as \(n \to \infty\) to find the radius of convergence \(R\).

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주요 개념

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Radius of Convergence

The radius of convergence of a power series is the distance from the center of the series within which the series converges absolutely. It determines the interval on the x-axis where the series sums to a finite value. Finding this radius helps understand the domain of validity for the series representation.

Ratio Test for Series Convergence

The ratio test is a method to determine the convergence of an infinite series by examining the limit of the absolute value of the ratio of consecutive terms. For power series, it is commonly used to find the radius of convergence by setting the limit of |a_{n+1}/a_n| equal to 1/|x|.

Stirling's Approximation

Stirling's approximation provides an estimate for factorials, especially useful for large n, stating n! ≈ √(2πn) (n/e)^n. This approximation simplifies factorial expressions in series terms, making it easier to analyze limits and apply convergence tests.

In Exercises 37–42, find the series’ radius of convergence.∑ (from n = 1 to ∞) [ n! xⁿ / nⁿ ]