5.2 Monotonic and bounded sequences

Monotonic and bounded sequences

Lessons

Notes:
Note *Theorem

1. A sequence is increasing if ana_n < an+1a_{n+1} for every n1n \geq 1.
2. A sequence is decreasing if ana_n > an+1a_{n+1} for every n1n \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 anNa_n \leq N for every n1n \geq 1.
5. A sequence is bounded below if there exists a number M such that anMa_n \geq M for every n1n \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.
  • 1.
    Overview:
  • 2.
    Difference between monotonic and non-monotonic sequences

    Show that the following sequences is monotonic. Is it an increasing or decreasing sequence?
  • 3.
    Difference between bounded, bounded above, and bounded below

    Determine whether the sequences are bounded below, bounded above, both, or neither
  • 4.
    Convegence of sequences

    Are the following sequences convergent according to theorem 7?
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Monotonic and bounded sequences

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