11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)5¹⁻²ᵏ

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Identify the given series: \( \sum_{k=1}^{\infty} 5^{1 - 2k} \). Notice that the general term can be rewritten to better understand its form.

Rewrite the term \( 5^{1 - 2k} \) as \( 5^{1} \cdot 5^{-2k} = 5 \cdot (5^{-2})^{k} \). This shows the series is a geometric series with common ratio \( r = 5^{-2} = \frac{1}{25} \).

Recall the convergence criterion for a geometric series: it converges if and only if \( |r| < 1 \). Since \( \frac{1}{25} < 1 \), the series converges.

To justify convergence, state that because the absolute value of the common ratio is less than 1, the infinite sum converges to \( \frac{a}{1 - r} \), where \( a \) is the first term of the series.

Find the first term \( a = 5^{1 - 2(1)} = 5^{-1} = \frac{1}{5} \) and express the sum formula for the series as \( S = \frac{a}{1 - r} = \frac{\frac{1}{5}}{1 - \frac{1}{25}} \), which confirms the series converges and allows calculation of its sum.

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주요 개념

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Geometric Series

A geometric series is a series where each term is obtained by multiplying the previous term by a constant ratio. It has the form ∑ ar^k, and its convergence depends on the absolute value of the ratio r. If |r| < 1, the series converges; otherwise, it diverges.

Convergence Tests

Convergence tests are methods used to determine whether an infinite series converges or diverges. For geometric series, the ratio test or direct recognition of the ratio can be applied. Understanding these tests helps justify the behavior of the series.

Exponential Expressions and Simplification

Simplifying exponential expressions is crucial to identify the common ratio in a series. Rewriting terms like 5^(1-2k) helps express the series in a standard geometric form, making it easier to analyze convergence.

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.∑ (from k = 1 to ∞)5¹⁻²ᵏ