 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-1/7/d/ii/solution

a_{n} = [n / (n + α)]^{n^{2}}

for

n \geq 1

∣ a_{n} ∣^{1/ n} = (1 + \frac{α}{n})^{- n} ⟶ e^{- α} .

For example, taking logarithms gives

- n lo g (1 + α / n) \to - α

. Therefore

R = e^{α} .

Any convention for the isolated

n = 0

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