42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞)5ᵏ / 2²ᵏ⁺¹

검증된 단계별 안내

Identify the given series: \( \sum_{k=1}^{\infty} \frac{5^k}{2^{2k+1}} \). This is an infinite series where each term is \( a_k = \frac{5^k}{2^{2k+1}} \).

Rewrite the general term to simplify the expression. Notice that \( 2^{2k+1} = 2^{2k} \cdot 2^1 = 2 \cdot (2^2)^k = 2 \cdot 4^k \). So, \( a_k = \frac{5^k}{2 \cdot 4^k} = \frac{1}{2} \cdot \left( \frac{5}{4} \right)^k \).

Recognize that the series is a geometric series with common ratio \( r = \frac{5}{4} \) and first term \( a_1 = \frac{1}{2} \cdot \left( \frac{5}{4} \right)^1 \).

Recall the convergence criterion for a geometric series: it converges if and only if \( |r| < 1 \). Here, check the absolute value of the ratio \( |r| = \frac{5}{4} \).

Since \( |r| > 1 \), conclude that the geometric series diverges by the geometric series test.

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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 series like the one given.

Geometric Series

A geometric series has terms that multiply by a constant ratio each time. It converges if the absolute value of this ratio is less than one, and its sum can be found using a specific formula. Recognizing the given series as geometric simplifies the convergence test.

Convergence Tests

Convergence tests, such as the Ratio Test or Root Test, help determine if a series converges or diverges. These tests analyze the limit of term ratios or roots to conclude about convergence. Choosing an appropriate test is key to solving series problems efficiently.

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.∑ (from k = 1 to ∞)5ᵏ / 2²ᵏ⁺¹