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

31.∑ (k = 1 to ∞) 2^(–3k)

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Identify the first term \( a \) of the geometric series by substituting \( k = 1 \) into the general term \( 2^{-3k} \). This gives \( a = 2^{-3 \times 1} = 2^{-3} \).

Determine the common ratio \( r \) by finding the ratio of the term at \( k = 2 \) to the term at \( k = 1 \). Calculate \( r = \frac{2^{-3 \times 2}}{2^{-3 \times 1}} = \frac{2^{-6}}{2^{-3}} \).

Simplify the common ratio \( r \) using the properties of exponents: \( \frac{2^{-6}}{2^{-3}} = 2^{-6 + 3} = 2^{-3} \).

Check the convergence of the series by verifying if the absolute value of the common ratio \( |r| < 1 \). Since \( r = 2^{-3} \), evaluate whether this condition holds.

If the series converges, use the formula for the sum of an infinite geometric series: \[ S = \frac{a}{1 - r} \]. Substitute the values of \( a \) and \( r \) to express the sum.

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주요 개념

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Geometric Series

A geometric series is the sum of the terms of a geometric sequence, 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 Infinite Geometric Series

An infinite 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 sum of infinitely many terms approaches a finite value.

Sum Formula for Convergent Geometric Series

When an infinite 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 direct way to find the total sum without adding infinitely many terms.

21–42. Geometric series Evaluate each geometric series or state that it diverges. 31.∑ (k = 1 to ∞) 2^(–3k)