Sequence

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A sequence is an ordered list of terms. Sequences may be either finite or infinite.

Definition

A sequence of real numbers is simply a function $f : \mathbb{N} \rightarrow \mathbb{R}$. For instance, the function $f(x) = x^2$ defined on $\mathbb{N}$ corresponds to the sequence $X = (x_n) = (0, 1, 4, 9, 16, \ldots)$.

Convergence

Intuitively, a sequence converges if its terms approach a particular number.

Formally, a sequence $(x_n)$ of reals converges to $L \in \mathbb{R}$ if and only if for all positive reals $\epsilon$, there exists a positive integer $k$ such that for all integers $n \ge k$, we have $|x_n - L| < \epsilon$. If $(x_n)$ converges to $L$, $L$ is called the limit of $(x_n)$ and is written $\lim_{n \to \infty} x_n$. The statement that $(x_n)$ converges to $L$ can be written as $(x_n)\rightarrow L$.

A classic example of convergence would be to show that $1/n\to 0$ as $n\to \infty$.

Claim: $\lim_{n\to\infty}\frac{1}{n}=0$.

Proof: Let $\epsilon>0$ be arbitrary and choose $N>\frac{1}{\epsilon}$. Then for $n\ge N$ we see that

$n>\frac{1}{\epsilon}\implies \frac{1}{n}<\epsilon\implies \left|\frac{1}{n}-0\right|<\epsilon$

which proves that $|x_n-L|<\epsilon$, so $1/n\to 0$ as $n\to \infty$ $\square$

Monotone Sequences

Many significant sequences have their terms continually increasing, such as $(n^2)$, or continually decreasing, such as $(1/n)$. This motivates the following definitions:

A sequence $(p_n)$ of reals is said to be

  • increasing if $p_n\leq p_{n+1}$ for all $n\in\mathbb{N}$ and strictly increasing if $p_n<p_{n+1}$ for all $n\in\mathbb{N}$,
  • decreasing if $p_n\geq p_{n+1}$ for all $n\in\mathbb{N}$ and strictly decreasing if $p_n>p_{n+1}$ for all $n\in\mathbb{N}$,
  • monotone if it is either decreasing or increasing.

Resources

See Also

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