Sequences

Sequences are ordered lists of numbers that may be finite or infinite. When we study Series, they are the sums of the terms of the sequence.

A sequence may be described as a function that maps natural numbers to the real number system.

\[ \begin{align*} f: \mathbb{N} \to \mathbb{R} \quad & \text{leads to} \\ a_n = f(n)& \end{align*} \]

The sequence may start at any integer.

\[ \begin{align*} \{{2n}\}^\infty_{n=-3} = a_{-3},a_{-2}, ... = \\ -6, -4, -2, ... \end{align*} \]

Properties¶

Increasing or Decreasing¶

Will a sequence continually rise in value as you go to the next term, or will it get lower? We can examine the relative adjacent terms to try to discover whether a sequence is "increasing" or "decreasing". We can also perform a first derivative test and and examine the values there to see if they are +'ve or -'ve.

An increasing sequence should have terms such that:

\[a_n < a_{n+1} \]

A decreasing sequence should have terms such that:

\[a_n > a_{n+1} \]

Strategies¶

Use the first derivative test to inspect where the function is decreasing/increasing.

Compare the nth term to the n+1th term.

Compare the ratio of the n+1th term to the nth term which follows from. The setup below is for a decreasing examination. $$ a_n \ge a_{n+1} \ a_{n+1} \le a_n \ \frac{a_{n+1}}{a_n} \ge 1 $$

Monotonocity¶

A function will be monotonic if it has the property of entirely increasing or decreasing for all elements in the domain.

Bounding¶

A sequence may never surpass a certain value, or it may never be lower than a certain value, or it may be contained between two numbers. If any of those are true, then the sequence may be "bounded above", "bounded below", "bounded between", or "not bounded".

\[ \begin{align*} If a_n \leq M \quad &\textbf{Bounded Above}. \\ If a_n \geq m \quad &\textbf{Bounded Below}. \\ If m \leq a_n \leq M \quad &\textbf{Bounded}. \\ \end{align*} \]

Limit¶

Convergence¶

The property of convergence relates to a sequence as it tends to infinity and whether the value of $a_n$ seems to go to a certain stable value. If that is not the case, then the sequence diverges.