48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.
∑ (k = 2 to ∞) 1 / eᵏ

Verified step by step guidance

1. Identify the series given: \( \sum_{k=2}^{\infty} \frac{1}{e^k} \). This is a series where each term is \( \frac{1}{e^k} \).
2. Recognize that this is a geometric series because the terms can be written as \( \left( \frac{1}{e} \right)^k \), where the common ratio \( r = \frac{1}{e} \).
3. Recall the geometric series test: a geometric series \( \sum_{k=0}^{\infty} ar^k \) converges if and only if \( |r| < 1 \).
4. Check the value of \( r = \frac{1}{e} \). Since \( e \approx 2.718 \), \( \frac{1}{e} < 1 \), so the series converges by the geometric series test.
5. To find the sum of the series starting from \( k=2 \), use the formula for the sum of a geometric series starting at \( k=0 \): \( S = \frac{a}{1-r} \), then adjust for the starting index by subtracting the first terms as needed.

Key Concepts

Geometric Series

A geometric series is a series where each term is obtained by multiplying the previous term by a constant ratio r. It converges if the absolute value of r is less than 1, and its sum can be found using the formula S = a / (1 - r), where a is the first term.

Convergence Tests for Series

Convergence tests help determine whether an infinite series converges or diverges. Common tests include the Ratio Test, Root Test, and Comparison Test, which analyze the behavior of terms or ratios to conclude about the series' convergence.

Exponential Functions in Series

Exponential functions like e^k grow rapidly, and their reciprocals (1/e^k) decrease exponentially. Recognizing such terms helps apply convergence tests effectively, as series with terms decreasing exponentially often converge.