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

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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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