Date | May 2008 | Marks available | 6 | Reference code | 08M.3ca.hl.TZ1.1 |
Level | HL only | Paper | Paper 3 Calculus | Time zone | TZ1 |
Command term | Determine | Question number | 1 | Adapted from | N/A |
Question
Determine whether the series \(\sum\limits_{n = 1}^\infty {\frac{{{n^{10}}}}{{{{10}^n}}}} \) is convergent or divergent.
Markscheme
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]
Examiners report
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.