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

Verified step by step guidance

1. Rewrite the general term of the series to simplify it. The term is given by \( \frac{7^k + 11^k}{11^k} \). Split this into two separate fractions: \( \frac{7^k}{11^k} + \frac{11^k}{11^k} \).
2. Simplify each fraction: \( \frac{7^k}{11^k} = \left(\frac{7}{11}\right)^k \) and \( \frac{11^k}{11^k} = 1 \). So the general term becomes \( \left(\frac{7}{11}\right)^k + 1 \).
3. Analyze the behavior of the terms as \( k \to \infty \). Since \( \left(\frac{7}{11}\right)^k \to 0 \), the term approaches \( 1 \).
4. Recall the necessary condition for series convergence: if \( \lim_{k \to \infty} a_k \neq 0 \), then the series \( \sum a_k \) diverges. Here, the limit of the terms is 1, not 0.
5. Conclude that the series \( \sum_{k=1}^\infty \frac{7^k + 11^k}{11^k} \) diverges because its terms do not approach zero.

Key Concepts

Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. Determining convergence means checking if the sum approaches a finite limit as the number of terms grows. Understanding the behavior of the series' terms is essential to decide if the series converges or diverges.

Geometric Series

A geometric series has terms that form a constant ratio with the previous term. It converges if the absolute value of the common ratio is less than one, and its sum can be calculated using a specific formula. Recognizing geometric series helps simplify and analyze series involving exponential terms.

Convergence Tests (Comparison and Limit Tests)

Convergence tests like the Comparison Test and the Limit Test help determine if a series converges by comparing it to known series or analyzing the limit of term ratios. These tests are crucial when series are not straightforward geometric series but can be related to them.