Sequence

A sequence on a set $A$ is a function $u:\mathbb{Z}^{+}\rightarrow A$.

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

Increasing or Decreasing

A sequence $\big(u_n\big)$ is

• Increasing iff $u_n\ge u_m$ for $n>m$
• Decreasing iff $u_n\le u_m$ for $n>m$
• Monotone iff either increasing or decreasing
• Strictly increasing iff $u_n\gt u_m$ for $n>m$
• Strictly decreasing iff $u_n\lt u_m$ for $n>m$

Convergence

Converging

A sequence $\big(u_n\big)_{n=1}^{\infty}$ is converging (to $L\in\mathbb{R}$) iff: $\lim_{n\to \infty}{u_n} = L$

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 $\big(u_n\big)$ be increasing and bounded above. Then $\big(u_n\big)$ is converging (to $\sup\,\set{u_n}$).

Decreasing and Bounded Below

Let $\big(u_n\big)$ be decreasing and bounded below. Then $\big(u_n\big)$ is converging (to $\inf\,\set{u_n}$).

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

Newton’s Method of Finding Roots

Suppose $f$ is a function. To find its roots:

• Select a point $x_0$
• Draw a tangent at $x_0$
• Choose $x_1$ which is where the tangent meets $y=0$
• Continue this process repeatedly

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