Textbook Question
21–42. Geometric series Evaluate each geometric series or state that it diverges.
21.∑ (k = 0 to ∞) (1/4)ᵏ
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14. Sequences & Series
Series
2:05 minutesProblem 10.3.27
Textbook Question
21–42. Geometric series Evaluate each geometric series or state that it diverges.
27.1 + 1.01 + 1.01² + 1.01³ + ⋯
Verified step by step guidance1
Identify the first term \( a \) and the common ratio \( r \) of the geometric series. Here, the first term is \( a = 1 \) and the common ratio is \( r = 1.01 \).
Recall that a geometric series \( a + ar + ar^2 + ar^3 + \cdots \) converges if and only if the absolute value of the common ratio \( |r| < 1 \).
Check the value of \( |r| \). Since \( r = 1.01 \), which is greater than 1, the series does not satisfy the convergence condition.
Conclude that because \( |r| > 1 \), the geometric series diverges and does not have a finite sum.
Therefore, the series \( 1 + 1.01 + 1.01^2 + 1.01^3 + \cdots \) diverges and cannot be evaluated to a finite 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 a + ar + ar² + ar³ + ⋯, where a is the first term and r is the common ratio.
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Geometric Series
Convergence and Divergence of Geometric Series
A geometric series converges if the absolute value of the common ratio |r| is less than 1, meaning the terms get smaller and approach zero. If |r| ≥ 1, the series diverges and does not have a finite sum.
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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 the finite sum of infinitely many terms.
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