Textbook Question
45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) (3/4)ᵏ
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14. Sequences & Series
Convergence Tests
2:08 minutesProblem 11.4.65c
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. ∑ₖ₌₀∞ (ln 2)ᵏ/k! = 2
Verified step by step guidance1
Recognize that the series given is an infinite sum of the form \(\sum_{k=0}^\infty \frac{(\ln 2)^k}{k!}\), which resembles the Taylor series expansion of the exponential function \(e^x = \sum_{k=0}^\infty \frac{x^k}{k!}\).
Identify that in this series, \(x = \ln 2\), so the sum can be rewritten as \(\sum_{k=0}^\infty \frac{(\ln 2)^k}{k!} = e^{\ln 2}\).
Recall the property of exponentials and logarithms that \(e^{\ln a} = a\) for any positive \(a\), so \(e^{\ln 2} = 2\).
Conclude that the series converges to 2, which means the statement \(\sum_{k=0}^\infty \frac{(\ln 2)^k}{k!} = 2\) is true.
Therefore, the explanation is that the series is the exponential series evaluated at \(\ln 2\), and by the fundamental properties of logarithms and exponentials, the sum equals 2.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Infinite Series and Convergence
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