9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.

∑ (k = 1 to ∞) 4ᵏ / (5ᵏ − 3)

검증된 단계별 안내

Identify the given series: \( \sum_{k=1}^{\infty} \frac{4^{k}}{5^{k} - 3} \). We want to determine if this series converges using the Comparison Test or the Limit Comparison Test.

Consider the behavior of the terms for large \( k \). Since \( 5^{k} \) grows much faster than 3, the denominator \( 5^{k} - 3 \) behaves approximately like \( 5^{k} \) for large \( k \). So the terms behave like \( \frac{4^{k}}{5^{k}} \).

Simplify the approximate terms: \( \frac{4^{k}}{5^{k}} = \left( \frac{4}{5} \right)^{k} \). This is a geometric series with ratio \( r = \frac{4}{5} \), which is less than 1, so the geometric series \( \sum \left( \frac{4}{5} \right)^{k} \) converges.

Use the Limit Comparison Test by computing \( \lim_{k \to \infty} \frac{\frac{4^{k}}{5^{k} - 3}}{\left( \frac{4}{5} \right)^{k}} \). Simplify this limit to check if it is a finite positive number.

If the limit is finite and positive, then by the Limit Comparison Test, the original series converges or diverges together with the geometric series \( \sum \left( \frac{4}{5} \right)^{k} \). Since the geometric series converges, conclude that the original series converges.

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주요 개념

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Comparison Test

The Comparison Test determines the convergence of a series by comparing it to another series with known behavior. If the terms of the given series are less than or equal to the terms of a convergent series, then the given series also converges. Conversely, if the terms are greater than or equal to those of a divergent series, the given series diverges.

Limit Comparison Test

The Limit Comparison Test involves taking the limit of the ratio of the terms of two series. If this limit is a positive finite number, both series either converge or diverge together. This test is useful when direct comparison is difficult but the series have similar dominant behavior for large terms.

Geometric Series and Dominant Term Analysis

Understanding geometric series is essential since the given series involves terms like 4^k and 5^k. For large k, the dominant terms in numerator and denominator determine the behavior of the series. Recognizing that 4^k / 5^k behaves like (4/5)^k helps in comparing the series to a convergent geometric series.

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.∑ (k = 1 to ∞) 4ᵏ / (5ᵏ − 3)