 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/i/solution

For the positive terms

a_{n} = n^{4} / 3^{n}

, the consecutive ratio is

\frac{a _{n + 1}}{a _{n}} = \frac{1}{3} (1 + \frac{1}{n})^{4} ⟶ \frac{1}{3} < 1.

The ratio test therefore proves that the series converges absolutely. More explicitly, for all sufficiently large

n

the ratio is at most a fixed

q < 1

, so the tail is bounded by a convergent geometric series.

 New to topics? Read the docs here!