Determine whether the series $\sum\limits_{n = 1}^\infty  {\frac{{{n^{10}}}}{{{{10}^n}}}}$ is convergent or divergent.

Consider

$\frac{{{u_{n + 1}}}}{{{u_n}}} = \frac{{{{(n + 1)}^{10}}}}{{{{10}^{n + 1}}}} \times \frac{{{{10}^n}}}{{{n^{10}}}}$     M1A1

$= \frac{1}{{10}}{\left( {1 + \frac{1}{n}} \right)^{10}}$     A1

$\to \frac{1}{{10}}{\text{ as }}n \to \infty$     A1

$\frac{1}{{10}} < 1$     R1

So by the Ratio Test the series is convergent.     R1

[6 marks]

Most candidates used the Ratio Test successfully to establish convergence. Candidates who attempted to use Cauchy’s (Root) Test were often less successful although this was a valid method.