Textbook Question
21–42. Geometric series Evaluate each geometric series or state that it diverges.
25.∑ (k = 0 to ∞) 0.9ᵏ
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14. Sequences & Series
Series
2:46 minutesProblem 10.3.33
Textbook Question
21–42. Geometric series Evaluate each geometric series or state that it diverges.
33.∑ (k = 4 to ∞) 1 / 5ᵏ
Verified step by step guidance1
Identify the first term \( a \) of the geometric series by substituting the starting index \( k = 4 \) into the term \( \frac{1}{5^k} \). So, \( a = \frac{1}{5^4} \).
Determine the common ratio \( r \) of the geometric series. Since the terms are of the form \( \frac{1}{5^k} \), the ratio between consecutive terms is \( \frac{1}{5} \).
Check the convergence of the series by verifying if the absolute value of the common ratio \( |r| < 1 \). If this condition holds, the series converges; otherwise, it diverges.
If the series converges, use the formula for the sum of an infinite geometric series starting at \( k = 0 \): \[ S = \frac{a}{1 - r} \]. Since our series starts at \( k = 4 \), \( a \) is already the first term at \( k=4 \).
Substitute the values of \( a \) and \( r \) into the sum formula to express the sum of the series. This will give the sum in terms of powers of 5 without calculating the final numeric value.
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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 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 helps in identifying and evaluating the series.
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Convergence of Infinite Geometric Series
An infinite geometric series converges if the absolute value of the common ratio |r| is less than 1. When it converges, the sum can be calculated using the formula S = a / (1 - r). If |r| ≥ 1, the series diverges and does not have a finite sum.
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Index Shift in Series Summation
When a series starts at an index other than zero or one, it may be necessary to adjust the formula for the sum by rewriting the series with a shifted index. This helps in correctly identifying the first term a and applying the geometric series sum formula.
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