Textbook Question
21–42. Geometric series Evaluate each geometric series or state that it diverges.
23.∑ (k = 0 to ∞) (–9/10)ᵏ
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14. Sequences & Series
Series
2:44 minutesProblem 10.3.31
Textbook Question
21–42. Geometric series Evaluate each geometric series or state that it diverges.
31.∑ (k = 1 to ∞) 2^(–3k)
Verified step by step guidance1
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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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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