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

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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.

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.

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 \( |x| < 1 \).

Check the convergence criteria: since \( \left| \frac{1}{6} \right| < 1 \) and \( \left| \frac{1}{3} \right| < 1 \), both series converge absolutely.

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.

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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.

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Convergence of an Infinite Series

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.

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Geometric Series

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.

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