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Monotonic and bounded sequences

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Intros
Lessons
  1. Overview:
  2. Monotonic Sequences
  3. Bounded Sequences
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Examples
Lessons
  1. Difference between monotonic and non-monotonic sequences

    Show that the following sequences is monotonic. Is it an increasing or decreasing sequence?
    1. {n2 n^2 }
    2. an=13na_n= \frac{1}{3^n}
    3. {nn+1}n=1∞ \{\frac{n}{n+1}\}_{n=1}^{\infty}
    4. {1, 1.5, 2, 2.5, 3, 3.5, ...}
  2. Difference between bounded, bounded above, and bounded below

    Determine whether the sequences are bounded below, bounded above, both, or neither
    1. an=n(−1)na_n=n(-1)^n
    2. an=(−1)nn2a_n=\frac{(-1)^n}{n^2}
    3. an=n3a_n=n^3
    4. an=−n4 a_n=-n^4
  3. Convegence of sequences

    Are the following sequences convergent according to theorem 7?
    1. {3n3}n=1∞ \{\frac{3}{n^3}\}_{n=1}^{\infty}
    2. {(−1)2n+12}n=1∞ \{\frac{(-1)^{2n+1}}{2}\}_{n=1}^{\infty}
    3. {n}n=4∞ \{\sqrt{n}\}_{n=4}^{\infty}
Topic Notes
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In this section, we will be talking about monotonic and bounded sequences. We will learn that monotonic sequences are sequences which constantly increase or constantly decrease. We also learn that a sequence is bounded above if the sequence has a maximum value, and is bounded below if the sequence has a minimum value. Of course, sequences can be both bounded above and below. Lastly, we will take a look at applying theorem 7, which will help us determine if the sequence is convergent. One important to note from the theorem is that even if theorem 7 does not apply to the sequence, there is a possibility that the sequence is convergent. It's just that the theorem will not be able to show it.
Note

Theorems:
1. A sequence is increasing if ana_n < an+1a_{n+1} for every n≥1n \geq 1.
2. A sequence is decreasing if ana_n > an+1a_{n+1} for every n≥1n \geq 1.
3. If a sequence is increasing or decreasing, then we call it monotonic.
4. A sequence is bounded above if there exists a number N such that an≤Na_n \leq N for every n≥1n \geq 1.
5. A sequence is bounded below if there exists a number M such that an≥Ma_n \geq M for every n≥1n \geq 1.
6. A sequence is bounded if it is both bounded above and bounded below.
7. If the sequence is both monotonic and bounded, then it is always convergent.
Basic Concepts
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