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: $$

$$\Big\{u_n\Big\}_{n=1}^{\infty} \;\; \text{or} \;\; \Big\{u_n\Big\} \;\; \text{or} \;\; \Big(u_n\Big)_{n=1}^{\infty}$$

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: $\lim_{n\to \infty}{u_n} = L$

$$\forall \epsilon > 0\; \exists N\in\mathbb{Z}^{+} \; \forall n \; ( n > N \implies |u_n-L| < \epsilon )$$

Diverging

A sequence is diverging iff it is not converging.

$$\lim_{n\to\infty}u_n= \begin{cases} \;\infty\\ \;-\infty\\ \;\text{undefined}, & \text{when}\;u_n\;\text{is osciallating}\\ \end{cases}$$

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 $\sup\,\set{u_n}$).

Proof Hint

• $\set{u_n}$ has a $\sup u_n (= s)$
• Prove: $\lim_{n\to\infty}u_n=s^{-}$

Decreasing and bounded below

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

Proof Hint

• $\set{u_n}$ has a $\inf u_n (= l)$
• Prove: $\lim_{n\to\infty}u_n=l^{+}$

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

$$x_{n+1}=x_n - \frac{f(x_n)}{f'(x_n)}$$

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