Sequence
A sequence on a set
Image of the n is written as
Increasing or Decreasing
A sequence
- Increasing iff
for - Decreasing iff
for - Monotone iff either increasing or decreasing
- Strictly increasing iff
for - Strictly decreasing iff
for
Convergence
Converging
A sequence
Diverging
A sequence is diverging iff it is not converging.
Convergence test
All converging sequences are bounded.
Increasing and bounded above
Let
Decreasing and bounded below
Let
Newton’s method of finding roots
Suppose
- Select a point
- Draw a tangent at
- Choose
which is where the tangent meets - Continue this process repeatedly