Introduction to Numerical Sequences and Real Analysis

Vocabulary practice flashcards covering fundamental concepts, definitions, graphical representations, and limit theorems of numerical sequences.

Numerical Sequence

A mathematical function whose domain is a subset $E$ of the natural numbers $\mathbb{N}$ containing all integers greater than or equal to a given non-negative integer $m$, and whose range is a subset of the real numbers $\mathbb{R}$.

General Term ($a_n$)

The $n$-th term of a sequence, representing the expression or formula that produces each value in the sequence corresponding to an index $n$.

Explicit Definition

A method of specifying a sequence in which the general term $a_n$ is given directly as a closed-form formula in terms of the independent index variable $n$.

Recurrence Definition

A method of defining a sequence by giving one or more initial terms along with a recurrence relation that expresses each subsequent term as a function of the preceding term or terms.

Periodic Sequence

A sequence for which there exists a positive integer period $P$ such that $a_{n+P} = a_n$ for all $n$ in the domain, causing the sequence values to repeat indefinitely every $P$ terms.

Fibonacci Sequence

A sequence defined by the recurrence relation $a_n = a_{n-1} + a_{n-2}$ with initial terms $a_1 = 1$ and $a_2 = 1$, historically modeled on the growth of pairs of rabbits and linked to the golden ratio.

Associated Function

A continuous real-valued function $f(x)$ whose rule coincides with a sequence such that $f(n) = a_n$, utilized to analyze sequence trends, construct plots, and evaluate limits.

First Bisector ($y = x$)

The identity line used in cobweb plots of recursively defined sequences ($u_{n+1} = f(u_n)$) to reflect values from the vertical axis back onto the horizontal axis to iteratively construct successive terms.

Convergent Sequence

A sequence whose general term $a_n$ approaches a single, finite real number $L$ as $n$ approaches positive infinity ($\lim_{n \to \infty} a_n = L$).

Divergent Sequence

A sequence that does not possess a finite real limit as $n$ tends to infinity, either because its values grow infinitely ($\pm\infty$) or oscillate without settling.

Squeeze (Sandwich) Theorem

A theorem stating that if a sequence $b_n$ is bounded between two sequences $a_n$ and $c_n$ ($a_n \le b_n \le c_n$) and both outer sequences converge to the same limit $L$, then the middle sequence $b_n$ must also converge to $L$.

Absolute Value Limit Theorem

A theorem establishing that if the limit of the absolute value of the terms is zero ($\lim_{n \to \infty} |a_n| = 0$), then the limit of the sequence itself must be zero ($\lim_{n \to \infty} a_n = 0$).

Composite Function Limit Theorem

A property stating that if $\lim_{n \to \infty} a_n = L$ and $f$ is a function continuous at $L$, then the limit of the sequence evaluated through $f$ is given by $\lim_{n \to \infty} f(a_n) = f(L)$.

Alternating Sign Factor

An expression such as $(-1)^n$ or $(-1)^{n+1}$ incorporated into the general term of a sequence to switch the sign between positive and negative for successive integer values of $n$.