Infinite Sequences

We start this review with infinite sequences. Recall that a sequence is an ordered list of numbers. We are interested in sequences where there is some rule or formula for the $n$th term in the sequence. For example,

$$\{a_n\}^{\infty}_{n=0}\;\;$$where$$\;\;a_n=(-1)^n\displaystyle \frac{n}{\sin(n)}$$

Then the 100th term in the list is $100/\sin(100)$. The 51st term in the list is $-51/\sin(51)$.

We will be interested to know what happens to the terms in the sequence as $n$ gets very large; i.e., what is $\lim_{n\to\infty}a_n$? To determine the limit of a sequence we need to realize that a sequence can be thought of as points on a continuous curve. For example, the sequence $\{1/n\}^{\infty}_{n=1}$ is simply points on the continuous function $f(x)=1/x$. Therefore the limit of the sequence is the limit of the function as $x$ gets large.

If the series is alternating as in the above example then half of the terms are on a positive continuous curve and the other half are on the negative counterpart of that continuous curve. In this case, the sequence limit will exist if and only if the limit of both continuous functions as $x$ gets large is zero. For example, the sequence $\{(-1)^n/n\}^{\infty}_{n=1}$ lies in the two curves $f(x)=1/x$ and $g(x)=-1/x$. Since $\lim_{x\to\infty}f(x)=\lim_{x\to\infty}g(x)=0$ the sequence converges to zero.

We have a few major results and definitions.

Definition 1   We say that a sequence is monotonic if either

1. $a_n\le a_{n+1}$ for all $n$. In this case we say the sequence is montonically increasing.

2. $a_n\ge a_{n+1}$ for all $n$. In this case we say the sequence is montonically decreasing.

Definition 2   We say that a sequence is bounded if there are numbers $M$ and $m$ such that $a_n\le M$ for all $n$ and $a_n\ge m$ for all $n$.

Theorem 1   Every bounded monotonic sequence converges.