42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞)2ᵏ / eᵏ

검증된 단계별 안내

Identify the given series: \( \sum_{k=1}^{\infty} \frac{2^k}{e^k} \). This is a series where each term is \( \frac{2^k}{e^k} \).

Rewrite the general term to recognize the type of series: \( \frac{2^k}{e^k} = \left( \frac{2}{e} \right)^k \). This shows the series is geometric with common ratio \( r = \frac{2}{e} \).

Recall the convergence criterion for a geometric series: A geometric series \( \sum r^k \) converges if and only if \( |r| < 1 \).

Evaluate the absolute value of the common ratio: \( \left| \frac{2}{e} \right| \). Since \( e \approx 2.718 \), compare \( 2 \) and \( e \) to determine if \( |r| < 1 \).

Based on the comparison, conclude whether the series converges or diverges by applying the geometric series test.

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Infinite Series

An infinite series is the sum of infinitely many terms, often expressed as ∑ a_k from k=1 to ∞. Understanding whether such a series converges (approaches a finite limit) or diverges (grows without bound or oscillates) is fundamental in calculus.

Geometric Series and Ratio Test

A geometric series has terms of the form ar^k. The Ratio Test compares the limit of |a_(k+1)/a_k| to 1; if less than 1, the series converges absolutely. This test is especially useful for series with exponential terms like 2^k/e^k.

Exponential Functions and Their Growth Rates

Exponential functions like 2^k and e^k grow at different rates. Since e ≈ 2.718, e^k grows faster than 2^k, which affects the behavior of the terms and helps determine convergence by comparing numerator and denominator growth.

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.∑ (from k = 1 to ∞)2ᵏ / eᵏ