72–86. Evaluating series Evaluate each series or state that it diverges.
∑ (k = 1 to ∞) (((1/6)ᵏ + (1/3)ᵏ) × k⁻¹)

Verified step by step guidance

1. Identify the given series: \( \sum_{k=1}^{\infty} \left( \left( \frac{1}{6} \right)^k + \left( \frac{1}{3} \right)^k \right) \times \frac{1}{k} \). This is a sum of two series combined inside the summation.
2. Rewrite the series by separating the sum into two separate series: \( \sum_{k=1}^{\infty} \frac{\left( \frac{1}{6} \right)^k}{k} + \sum_{k=1}^{\infty} \frac{\left( \frac{1}{3} \right)^k}{k} \). This allows us to analyze each series individually.
3. Recognize that each series is of the form \( \sum_{k=1}^{\infty} \frac{x^k}{k} \), which is related to the Taylor series expansion of the function \( -\ln(1 - x) \) for \( |x| < 1 \).
4. Check the convergence criteria: since \( \left| \frac{1}{6} \right| < 1 \) and \( \left| \frac{1}{3} \right| < 1 \), both series converge absolutely.
5. Express each series using the logarithmic function: \( \sum_{k=1}^{\infty} \frac{x^k}{k} = -\ln(1 - x) \). Substitute \( x = \frac{1}{6} \) and \( x = \frac{1}{3} \) respectively, then add the two results to write the sum in closed form.

Key Concepts

Convergence and Divergence of Infinite Series

An infinite series converges if the sum of its terms approaches a finite limit as the number of terms grows indefinitely. Otherwise, it diverges. Determining convergence is essential before evaluating the sum, often using tests like the comparison, ratio, or root tests.

p-Series and Geometric Series

A p-series has terms of the form 1/k^p and converges if p > 1. A geometric series has terms of the form ar^k and converges if |r| < 1. The given series combines these forms, so understanding their convergence criteria helps analyze the overall behavior.

Term-by-Term Analysis and Limit Comparison Test

When a series is a sum of multiple sequences, analyzing each part separately can clarify convergence. The Limit Comparison Test compares terms of the given series to a known benchmark series to determine convergence or divergence, especially useful for series with combined terms.