 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-1/19/a/iii/solution

The summands repeat the six-term pattern

\frac{3}{2}, \frac{3}{2}, 0, - \frac{3}{2}, - \frac{3}{2}, 0.

In particular they do not tend to zero: the subsequence with

n = 6 k + 1

is always

3 /2

. The term test for divergence therefore shows that the series diverges. Although each block of six sums to zero, that only makes a subsequence of partial sums zero; it does not make the full sequence of partial sums convergent.

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