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 ∞) k¹⁰⁰ / (k + 1)!

Verified step by step guidance

1. Identify the given series: \( \sum_{k=1}^{\infty} \frac{k^{100}}{(k+1)!} \). We want to determine if it converges absolutely, conditionally, or diverges.
2. Since the terms involve factorials, consider using the Ratio Test, which is effective for series with factorial expressions.
3. Set up the Ratio Test by examining the limit \( L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \), where \( a_k = \frac{k^{100}}{(k+1)!} \). Write \( a_{k+1} = \frac{(k+1)^{100}}{(k+2)!} \).
4. Compute the ratio \( \frac{a_{k+1}}{a_k} = \frac{(k+1)^{100}}{(k+2)!} \times \frac{(k+1)!}{k^{100}} \). Simplify the factorial terms and the powers to express the ratio in a form suitable for taking the limit as \( k \to \infty \).
5. Evaluate the limit \( L \). If \( L < 1 \), the series converges absolutely; if \( L > 1 \), it diverges; if \( L = 1 \), the test is inconclusive and another test must be used.

Key Concepts

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 this limit is less than 1, the series converges absolutely; if greater than 1, it diverges. It is especially useful for series involving factorials.

Factorials and Growth Rates

Factorials grow faster than any polynomial function. Recognizing how factorial terms dominate polynomial terms in the numerator or denominator helps predict the behavior of series and guides the choice of convergence tests.