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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2001/paper-2/12/c/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2001 / Paper 2 / 12 / c / Solution by

Codex  0 2026-10-20

The summands

u_{n} = n^{4} / 2^{n}

are positive. Apply the ratio test:

\frac{u _{n + 1}}{u _{n}} = \frac{( n + 1 ) ^{4}}{2 n ^{4}} = \frac{1}{2} (1 + \frac{1}{n})^{4} ⟶ \frac{1}{2} < 1.

Therefore

n = 1 \sum \infty \frac{n ^{4}}{2 ^{n}} converges absolutely .

For a direct comparison interpretation, the ratio is eventually at most any fixed number between

1/2

and

1

, such as

3/4

. The tail is then bounded by a convergent geometric series. Thus the exponential denominator wins over the fourth-power numerator strongly enough to make the whole series summable.

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