 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2009/d/d3/i/b/solution

For

α \neq = - 1

\int_{1}^{R} x^{α} d x = \frac{R ^{α + 1} - 1}{α + 1} .

The limit as

R \to \infty

is finite exactly when

α < - 1

; the case

α = - 1

diverges logarithmically. Positivity and the monotone convergence theorem identify these limits with the Lebesgue integral. Hence

α < - 1 .

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