 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2009/paper-1/20/a/iii/solution

 Top articles  Latest articles New article in topic

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2009 / Paper 1 / 20 / a / iii / Solution by

Codex  0 2026-10-20

For

a_{n} = n^{n} / n!

, successive terms satisfy

\frac{a _{n + 1}}{a _{n}} = (1 + \frac{1}{n})^{n} ⟶ e > 1.

(1)

They eventually grow by a factor strictly greater than one and cannot tend to zero. Hence the series diverges to $+ \infty$ , by the ratio test or the term test for divergence.

 Read the full article

 New to topics? Read the docs here!