 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-1/19/c/ii/solution

Compare the positive terms with those of the harmonic series:

\frac{( n ^{2} + 2 n ) / ( n ^{3} + 3 n ^{2} + 1 )}{1/ n} = \frac{n ^{3} + 2 n ^{2}}{n ^{3} + 3 n ^{2} + 1} ⟶ 1.

By the limit comparison test, the given series has the same convergence behaviour as

\sum 1/ n

, which diverges. Thus the series diverges to $+ \infty$ .

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