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Exercise 2: Monotonicity

Theory

Determine whether each sequence is monotone. If monotone, state whether it is increasing or decreasing.

Exercises

Exercise 1
Question
{n2n+1}\left\{ \frac{n}{2n+1} \right\}
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Solution

Method: Difference.

an+1−an=n+12n+3−n2n+1=(n+1)(2n+1)−n(2n+3)(2n+3)(2n+1).a_{n+1} - a_n = \frac{n+1}{2n+3} - \frac{n}{2n+1} = \frac{(n+1)(2n+1) - n(2n+3)}{(2n+3)(2n+1)}.= \frac{2n^2+3n+1 - 2n^2-3n}{(2n+3)(2n+1)} = \frac{1}{(2n+3)(2n+1)} > 0.

Conclusion: The sequence is monotone increasing.

Exercise 2
Question
{n−2n}\left\{ n - 2^n \right\}
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Solution

Method: Difference.

an+1−an=[(n+1)−2n+1]−[n−2n]=1−2n.a_{n+1} - a_n = [(n+1)-2^{n+1}] - [n-2^n] = 1 - 2^n.

For n≥1n \ge 1, 2^n > 1, so

a_{n+1} - a_n < 0.

Conclusion: The sequence is monotone decreasing.

Exercise 3
Question
{n!3n}\left\{ \frac{n!}{3^n} \right\}
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Solution

Method: Ratio.

an+1an=(n+1)!/3n+1n!/3n=n+13.\frac{a_{n+1}}{a_n} = \frac{(n+1)!/3^{n+1}}{n!/3^n} = \frac{n+1}{3}.

For n≥3n \ge 3, \frac{n+1}{3} > 1, while for small nn, the ratio is <1.

Conclusion: The sequence decreases initially and then increases, so it is not monotone.