Monotonic and bounded sequences

Monotonic and bounded sequences

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.
Basic concepts: Introduction to sequences,

Lessons

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.
  • Introduction
    Overview:
    a)
    Monotonic Sequences

    b)
    Bounded Sequences

  • 1.
    Difference between monotonic and non-monotonic sequences

    Show that the following sequences is monotonic. Is it an increasing or decreasing sequence?
    a)
    {n2 n^2 }

    b)
    an=13na_n= \frac{1}{3^n}

    c)
    {nn+1}n=1∞ \{\frac{n}{n+1}\}_{n=1}^{\infty}

    d)
    {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
    a)
    an=n(−1)na_n=n(-1)^n

    b)
    an=(−1)nn2a_n=\frac{(-1)^n}{n^2}

    c)
    an=n3a_n=n^3

    d)
    an=−n4 a_n=-n^4

  • 3.
    Convegence of sequences

    Are the following sequences convergent according to theorem 7?
    a)
    {3n3}n=1∞ \{\frac{3}{n^3}\}_{n=1}^{\infty}

    b)
    {(−1)2n+12}n=1∞ \{\frac{(-1)^{2n+1}}{2}\}_{n=1}^{\infty}

    c)
    {n}n=4∞ \{\sqrt{n}\}_{n=4}^{\infty}

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5.
Sequence and Series
5.1
Introduction to sequences
5.2
Monotonic and bounded sequences
5.3
Introduction to infinite series
5.4
Convergence and divergence of normal infinite series
5.5
Convergence & divergence of geometric series
5.6
Convergence & divergence of telescoping series
5.7
Divergence of harmonic series
5.8
P Series
5.9
Alternating series test
5.10
Divergence test
5.11
Comparison & limit comparison test
5.12
Integral test
5.13
Ratio test
5.14
Root test
5.15
Absolute & conditional convergence
5.16
Radius and interval of convergence with power series
5.17
Functions expressed as power series
5.18
Taylor series and Maclaurin series
5.19
Approximating functions with Taylor polynomials and error bounds
Calculus 2
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