11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (−7)ᵏ / k!

Verified step by step guidance

1. Identify the given series: \( \sum_{k=1}^{\infty} \frac{(-7)^k}{k!} \). This is an infinite series with terms involving factorials in the denominator.
2. Recall that factorials grow very rapidly, which often suggests the series might converge. To confirm, consider applying the Ratio Test, which is useful for series with factorials and exponentials.
3. Set up the Ratio Test by examining the limit \( L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \), where \( a_k = \frac{(-7)^k}{k!} \). Substitute to get \( L = \lim_{k \to \infty} \left| \frac{(-7)^{k+1} / (k+1)!}{(-7)^k / k!} \right| \).
4. Simplify the expression inside the limit: \( L = \lim_{k \to \infty} \left| \frac{-7}{k+1} \right| = \lim_{k \to \infty} \frac{7}{k+1} \). Since \( \frac{7}{k+1} \to 0 \) as \( k \to \infty \), the limit \( L = 0 \).
5. Interpret the Ratio Test result: since \( L < 1 \), the series \( \sum_{k=1}^{\infty} \frac{(-7)^k}{k!} \) converges absolutely. Therefore, the series converges.

Key Concepts

Convergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. Understanding convergence is essential to determine whether the sum of infinitely many terms results in a finite value or diverges to infinity or oscillates.

Ratio Test

The ratio test evaluates the limit of the absolute value of the ratio of consecutive terms. If this limit is less than one, the series converges absolutely; if greater than one, it diverges. This test is particularly useful for series involving factorials or exponential terms.

Factorials and Their Growth

Factorials (n!) grow very rapidly compared to exponential or polynomial terms. Recognizing the dominance of factorial growth helps in applying convergence tests, as terms with factorials in the denominator often lead to convergence due to rapid term size reduction.