11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) 2ᵏ / (3ᵏ − 2ᵏ)

검증된 단계별 안내

Identify the given series: \( \sum_{k=1}^{\infty} \frac{2^{k}}{3^{k} - 2^{k}} \). We want to determine if this series converges or diverges.

Analyze the general term \( a_k = \frac{2^{k}}{3^{k} - 2^{k}} \). For large \( k \), compare the dominant terms in the denominator to simplify the expression.

Since \( 3^{k} \) grows faster than \( 2^{k} \), for large \( k \), \( 3^{k} - 2^{k} \approx 3^{k} \). So, \( a_k \approx \frac{2^{k}}{3^{k}} = \left( \frac{2}{3} \right)^{k} \).

Use the Comparison Test or Limit Comparison Test by comparing \( a_k \) with the geometric series \( \sum \left( \frac{2}{3} \right)^{k} \), which is a convergent geometric series because \( \left| \frac{2}{3} \right| < 1 \).

Conclude that since \( a_k \) behaves like a convergent geometric series for large \( k \), the original series \( \sum_{k=1}^{\infty} \frac{2^{k}}{3^{k} - 2^{k}} \) converges by the Comparison Test.

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주요 개념

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Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. Determining whether such a series converges means checking if the sum approaches a finite limit as the number of terms grows indefinitely. Understanding convergence is essential to analyze the behavior of the given series.

Comparison Test

The Comparison Test involves comparing the given series to a known benchmark series with established convergence properties. If the terms of the given series are smaller than those of a convergent series, it also converges; if larger than a divergent series, it diverges. This test helps in establishing convergence by bounding the series.

Ratio Test

The Ratio Test examines the limit of the ratio of consecutive terms in a series. 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 exponential terms, like powers of k, as in the given problem.

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.∑ (from k = 1 to ∞) 2ᵏ / (3ᵏ − 2ᵏ)