Consider the infinite series \(\sum\limits_{n = 1}^\infty  {\frac{2}{{{n^2} + 3n}}} \).

Use a comparison test to show that the series converges.

EITHER

\(\sum\limits_{n = 1}^\infty  {\frac{2}{{{n^2} + 3n}}}  < \sum\limits_{n = 1}^\infty  {\frac{2}{{{n^2}}}} \)     M1

which is convergent     A1

the given series is therefore convergent using the comparison test     AG

OR

\(\mathop {{\text{lim}}}\limits_{n \to \infty } \frac{{\frac{2}{{{n^2} + 3n}}}}{{\frac{1}{{{n^2}}}}} = 2\)     M1A1

the given series is therefore convergent using the limit comparison test     AG

[2 marks]

Most candidates were able to answer part (a) and many gained a fully correct answer. A number of candidates ignored the factor 2 in the numerator and this led to candidates being penalised. In some cases candidates were not able to identify an appropriate series to compare with. Most candidates used the Comparison test rather than the Limit comparison test.