Monotone and Bounded Sequences

Theory

A sequence is monotone if it is increasing, non-decreasing, decreasing, or non-increasing.
Common methods to test monotonicity:

• Difference: $a_{n+1} - a_n$
• Ratio: $\dfrac{a_{n+1}}{a_n}$
• Derivative of the related function $f(x)$

Every bounded monotone sequence is convergent.

Beginner Checking Pattern

1. Choose a method (difference, ratio, or derivative)
2. Determine increasing or decreasing
3. Check boundedness
4. Conclude convergence or divergence

Decision Tree: Monotone, Bounded, Convergent

1. Check for oscillation.
   If the sequence alternates signs or oscillates
   (e.g. $(-1)^n$, $\sin n$, $\cos(n\pi)$),
   then it is not monotone.
2. Test monotonicity.
   If possible, compute

$$a_{n+1}-a_n \quad \text{or} \quad \frac{a_{n+1}}{a_n}.$$

• Positive $\Rightarrow$ increasing
• Negative $\Rightarrow$ decreasing

3. Use derivatives if needed.
   Write $a_n = f(n)$ and compute $f'(x)$.

• f'(x) > 0 \Rightarrow increasing
• f'(x) < 0 \Rightarrow decreasing

4. Check boundedness.
   Find constants $m$ and $M$ such that

$$m \le a_n \le M.$$

5. Conclude.

• Monotone + bounded $\Rightarrow$ convergent
• Unbounded $\Rightarrow$ divergent

Exercises

Exercise 1

Question

Check whether

$$\left\{ 1, \frac{2^2}{2!}, \frac{2^3}{3!}, \frac{2^4}{4!}, \dots \right\}$$

is monotone.

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Solution

Method: Ratio.

\frac{a_{n+1}}{a_n} = \frac{2^{n+1}/(n+1)!}{2^n/n!} = \frac{2}{n+1} < 1 \quad \text{for } n \ge 2.

Conclusion: The sequence is decreasing (monotone decreasing).

Exercise 2

Question

Check whether

$$\left\{ n - \left(\frac12\right)^n \right\}$$

is monotone.

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Solution

Method: Difference.

a_{n+1} - a_n = \left[(n+1) - \left(\frac12\right)^{n+1}\right] - \left[n - \left(\frac12\right)^n\right] = 1 + \left(\frac12\right)^n\!\left(1 - \frac12\right) > 0.

Conclusion: The sequence is increasing (monotone increasing).

Exercise 3

Question

Is

$$\left\{ \cos(n\pi) \right\}$$

monotone, bounded, or convergent?

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Solution

Observation:

$$\cos(n\pi) = (-1)^n = 1, -1, 1, -1, \dots$$

Conclusion:

• Not monotone
• Bounded between $-1$ and $1$
• Not convergent

Exercise 4

Question

Is

$$\left\{ \frac{(-1)^{n+1} n}{n^2+1} \right\}$$

monotone, bounded, or convergent?

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Solution

Boundedness:

\left| \frac{(-1)^{n+1} n}{n^2+1} \right| \le \frac{n}{n^2+1} < 1.

Limit:

$$\lim_{n\to\infty} \frac{(-1)^{n+1} n}{n^2+1} = 0.$$

Conclusion:

• Not monotone (sign alternates)
• Bounded
• Convergent to $0$