21–42. Geometric series Evaluate each geometric series or state that it diverges.

41.∑ (k = 1 to ∞) 4 / 12ᵏ

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Identify the first term $ a $ and the common ratio $ r $ of the geometric series. Here, the series is $ \sum_{k=1}^{\infty} \frac{4}{12^k} $, so the first term is $ a = \frac{4}{12^1} = \frac{4}{12} $.

Express the series in the standard geometric series form $ \sum_{k=0}^{\infty} ar^k $ by adjusting the index if necessary. Since the sum starts at $ k=1 $, rewrite it as $ \sum_{k=0}^{\infty} ar^k $ with $ a = \frac{4}{12} $ and $ r = \frac{1}{12} $.

Check the convergence of the series by evaluating the absolute value of the common ratio $ |r| $. If $ |r| < 1 $, the series converges; otherwise, it diverges.

If the series converges, use the formula for the sum of an infinite geometric series starting at $ k=0 $: $\n\[\n$\$\$ S = $\frac{a}{1 - r}$ \$\$$\n\]\nwhere$ $ a $ is the first term and $ r $ is the common ratio.

Calculate the sum using the values of $ a $ and $ r $ identified, keeping the expression symbolic without simplifying to a final decimal value.

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주요 개념

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Geometric Series

A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio. It has the form ∑ ar^(k), where a is the first term and r is the common ratio. Understanding this structure is essential to evaluate or determine the convergence of the series.

Convergence Criteria for Geometric Series

A geometric series converges if and only if the absolute value of the common ratio |r| is less than 1. If |r| ≥ 1, the series diverges. This criterion helps decide whether the infinite sum has a finite value or not.

Sum Formula for Convergent Geometric Series

When a geometric series converges, its sum can be calculated using the formula S = a / (1 - r), where a is the first term and r is the common ratio. This formula provides a quick way to find the total sum of infinitely many terms.

21–42. Geometric series Evaluate each geometric series or state that it diverges. 41.∑ (k = 1 to ∞) 4 / 12ᵏ