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Sahithyan's S1

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.

Increasing and bounded above

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

Decreasing and bounded below

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

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