 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/ii/solution

Add the supplied five-term partial sum to each side of the tail inequality. Since

1/6 = 600/3600

and

1/5 = 720/3600

\frac{5869}{3600} < n = 1 \sum \infty \frac{1}{n ^{2}} < \frac{5989}{3600} .

Numerically this is approximately

1.63028 < \sum n^{- 2} < 1.66361

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