 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-ii/1/c/ii/solution

The sine has values

1, 0, - 1, 0

on the four residue classes of

n

modulo four. For

n = 4 k + 1

b_{n} = e^{1 - 1/ n} \to e

; the other residue classes tend to

1, e^{- 1}, 1

. All terms satisfy

b_{n} \leq e

, and the first class approaches

e

. Hence

n \to \infty lim sup e^{(1 - 1/ n) s i n (nπ /2)} = e .

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