 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-ii/2/b/i/solution

The implication is true, with absolute convergence. Since

n^{3/2} a_{n} \to 0

, there is

N

such that

∣ a_{n} ∣ \leq n^{- 3/2}

for

n \geq N

\sum n^{- 3/2}

converges because its exponent exceeds one. The comparison test proves

\sum ∣ a_{n} ∣ < \infty

\sum a_{n}

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