 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2003/paper-1/10/a/i/solution

For

n \geq 6

and

n - 1 < x < n

, monotonicity gives

(x + 1)^{- 2} < n^{- 2} < x^{- 2}

. Integrating over each unit interval and summing proves the integral test tail bounds

\int_{5}^{\infty} \frac{d x}{( x + 1 ) ^{2}} = \frac{1}{6} < n = 6 \sum \infty \frac{1}{n ^{2}} < \int_{5}^{\infty} \frac{d x}{x ^{2}} = \frac{1}{5} .

The inequalities remain strict because the very first rectangle already has a strictly positive difference from either bounding integral.

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