In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑ (from n = 0 to ∞) [(2n + 3)(2ⁿ + 3) / (3ⁿ + 2)]

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First, write down the general term of the series: \(a_n = \frac{(2n + 3)(2^n + 3)}{3^n + 2}\).

To determine convergence or divergence, analyze the behavior of \(a_n\) as \(n\) approaches infinity. Focus on the dominant terms in the numerator and denominator for large \(n\).

Approximate the dominant terms: in the numerator, \((2n + 3)(2^n + 3) \approx 2n \cdot 2^n = 2n \cdot 2^n\), and in the denominator, \(3^n + 2 \approx 3^n\) for large \(n\).

Simplify the approximate form of \(a_n\) for large \(n\): \(a_n \approx \frac{2n \cdot 2^n}{3^n} = 2n \left(\frac{2}{3}\right)^n\).

Use the Divergence Test (also called the nth-term test): if \(\lim_{n \to \infty} a_n \neq 0\), the series diverges. Otherwise, consider applying a convergence test such as the Ratio Test or Root Test to confirm convergence.

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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 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 and Limit Comparison Tests

These tests help determine convergence by comparing the given series to a known benchmark series. The Comparison Test involves directly comparing terms, while the Limit Comparison Test uses the limit of the ratio of terms. They are useful when series involve exponential or polynomial expressions.

Ratio Test

The Ratio Test examines the limit of the absolute value of the ratio of consecutive terms. If this limit is less than one, the series converges absolutely; if greater than one, it diverges. This test is particularly effective for series with exponential terms like powers of n.

In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.∑ (from n = 0 to ∞) [(2n + 3)(2ⁿ + 3) / (3ⁿ + 2)]