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 $(x_n)\rightarrow L$.

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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