Sequences

A sequence on a set is a function .

Image of the n is written as . A sequence is indicated by one of these ways:

Convergence

Converging

A sequence is converging (to ) iff:

Note

Diverging

A sequence is diverging iff it is not converging.

Increasing or Decreasing

A sequence is

• Increasing iff for
• Decreasing iff for
• Monotone iff either increasing or decreasing
• Strictly increasing iff for
• Strictly decreasing iff for

Convergence test

Increasing and bounded above

Let be increasing and bounded above. Then is converging (to ).

Proof Hint

• has a
• Prove:

Decreasing and bounded below

Let be decreasing and bounded below. Then is converging (to ).

Proof Hint

• has a
• Prove:

Newton’s method of finding roots

Suppose is a function. To find its roots:

• Select a point
• Draw a tangent at
• Choose which is where the tangent meets
• Continue this process repeatedly