Monotonic and bounded sequences

Lessons

• 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)

  {$n^2$}
  
  
  b)

  $a_n= \frac{1}{3^n}$
  
  
  c)

  $\{\frac{n}{n+1}\}_{n=1}^{\infty}$
  
  
  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)

  $a_n=n(-1)^n$
  
  
  b)

  $a_n=\frac{(-1)^n}{n^2}$
  
  
  c)

  $a_n=n^3$
  
  
  d)

  $a_n=-n^4$
  
  

---

• 3.
  Convegence of sequences
  
  Are the following sequences convergent according to theorem 7?
  a)

  $\{\frac{3}{n^3}\}_{n=1}^{\infty}$
  
  
  b)

  $\{\frac{(-1)^{2n+1}}{2}\}_{n=1}^{\infty}$
  
  
  c)

  $\{\sqrt{n}\}_{n=4}^{\infty}$
  
  

---