Exponentials dominate powers in sequences

Builds on Limit of a sequence The sequence r^n tends to infinity for r > 1

Statement

Let $k$ be a natural number and let $r$ be a real number with $r > 1$ . Then $\lim_{n \to \infty} \frac{n^k}{r^n} = 0.$

Remarks

For $k = 1$ and $r = 2$ , it says $n / 2^n \to 0$ . The terms are $1/2, 2/4, 3/8, 4/16, \dots$ . One proof uses the ratio of consecutive terms. The ratio $\frac{(n+1)^k}{n^k} \cdot \frac{1}{r}$ tends to $1/r < 1$ , so the terms eventually shrink by a fixed factor. In the language of little o notation, it says $n^k = o(r^n)$ . The node exponential growth beats polynomial growth gives the version for real variables.

Reading path

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Sequence definition
Real sequence definition
Geometric sequence definition
Limit of a sequence definition
Sequence tending to infinity definition
The sequence r^n tends to infinity for r > 1 example
Exponentials dominate powers in sequences statement

Tag	`0MVTCFW42K`
URI	`m2://0MVTCFW42K`
Kind	statement, proposition
Writing	W0 stub
MSC	26A12
Status	This copy of the site has no build data. Run `m2 status 0MVTCFW42K`.
Uses	Limit of a sequence, The sequence r^n tends to infinity for r > 1

Exponentials dominate powers in sequences

Statement

Remarks

Reading path

Further reading

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