Textbook Question
21–42. Geometric series Evaluate each geometric series or state that it diverges.
27.1 + 1.01 + 1.01² + 1.01³ + ⋯
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14. Sequences & Series
Series
3:17 minutesProblem 10.3.35
Textbook Question
21–42. Geometric series Evaluate each geometric series or state that it diverges.
35.∑ (k = 0 to ∞) 3(–π)^(–k)
Verified step by step guidance1
Identify the first term \( a \) of the geometric series by substituting \( k = 0 \) into the general term \( 3(-\pi)^{-k} \). This gives \( a = 3(-\pi)^0 = 3 \).
Determine the common ratio \( r \) by finding the factor that each term is multiplied by to get the next term. This is \( r = \frac{3(-\pi)^{-1}}{3(-\pi)^0} = (-\pi)^{-1} = \frac{1}{-\pi} = -\frac{1}{\pi} \).
Check the convergence of the series by evaluating the absolute value of the common ratio \( |r| = \left| -\frac{1}{\pi} \right| = \frac{1}{\pi} \). Since \( \pi > 1 \), \( |r| < 1 \), so the series converges.
Use the formula for the sum of an infinite geometric series \( S = \frac{a}{1 - r} \) to express the sum, where \( a = 3 \) and \( r = -\frac{1}{\pi} \).
Write the sum explicitly as \( S = \frac{3}{1 - \left(-\frac{1}{\pi}\right)} = \frac{3}{1 + \frac{1}{\pi}} \). Simplify this expression to get the sum in a more compact form.
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Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Geometric Series Definition
A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio, r. It has the form ∑ ar^k, where a is the first term and k is the index. Understanding this structure is essential to identify and evaluate 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 determines whether the sum approaches a finite value or not.
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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 a quick way to find the total sum of infinitely many terms.
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