Sequence

A sequence on a set is a function .

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

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

Converging

A sequence is converging (to ) iff:

Diverging

A sequence is diverging iff it is not converging.

Convergence test

All converging sequences are bounded. Contrapositive can be used to prove the divergence.

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:

Both of the above results are referred to as “monotone convergence theorem”.

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

Sequence of converges to one of the roots. Different points can be taken to find other roots.