Textbook Question
21–42. Geometric series Evaluate each geometric series or state that it diverges.
29.∑ (k = 1 to ∞) e^(–2k)
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14. Sequences & Series
Series
1:43 minutesProblem 10.3.37
Textbook Question
21–42. Geometric series Evaluate each geometric series or state that it diverges.
37.1 + e/π + e²/π² + e³/π³ + ⋯
Verified step by step guidance1
Identify the first term \( a \) of the geometric series. Here, the first term is \( 1 \).
Determine the common ratio \( r \) by dividing the second term by the first term: \( r = \frac{e/\pi}{1} = \frac{e}{\pi} \).
Check the convergence of the series by evaluating the absolute value of the common ratio \( |r| = \left| \frac{e}{\pi} \right| \). The series converges if \( |r| < 1 \) and diverges otherwise.
If the series converges, use the formula for the sum of an infinite geometric series: \[ S = \frac{a}{1 - r} \], where \( a \) is the first term and \( r \) is the common ratio.
Substitute the values of \( a \) and \( r \) into the formula to express the sum of the series.
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Video duration:
1mKey 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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Convergence of Geometric Series
A 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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Evaluating the Given Series
To evaluate the series 1 + e/π + e²/π² + ⋯, identify the first term a = 1 and the common ratio r = e/π. Since e ≈ 2.718 and π ≈ 3.1415, |r| < 1, so the series converges. Use the sum formula S = 1 / (1 - e/π) to find the sum.
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