Sequences, Limits, and Asymptotics

Flashcards defining core concepts, theorems, classifications, asymptotic relations, and limit properties for real and vector sequences.

Sequence of Real Numbers

A function $f: \mathbb{N}^+ \rightarrow \mathbb{R}$ (where $\mathbb{N}^+ = \mathbb{N} \setminus \{0\}$) that associates to each natural number $n \ge 1$ a real number $f(n)$.

Terms of a Sequence

The real numbers that represent the image values of a sequence function, commonly denoted by $a_n$, $b_n$, or $x_n$.

Harmonic Sequence

The sequence defined on $\mathbb{N}^+$ by $a_n = \frac{1}{n}$, generating the list of terms $\{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots\}$.

Recurrence (Recursion)

A method of defining a sequence by specifying one or more initial values and a rule that determines each subsequent term from the preceding ones (e.g., $a_{n+1} = f(a_n)$).

Fibonacci Sequence

The sequence defined by the initial conditions $a_0 = 1$, $a_1 = 1$ and the recurrence relation $a_{n+2} = a_n + a_{n+1}$ for all $n \ge 0$, yielding the terms $\{1, 1, 2, 3, 5, 8, 13, 21, \dots\}$.

Arithmetic Sequence

A sequence characterized by a constant difference $b$ (or $d$) between successive terms, defined recursively by $a_1 = a$ and $a_n = a_{n-1} + b$ for $n \ge 2$, or in closed form by $a_n = a + dn$.

Geometric Sequence

A sequence defined recursively by $a_{n+1} = q a_n$ with $a_0 = a$, or in closed form by $a_n = a q^{n-1}$ (for $n \ge 1$), where $a$ is the first term and $q$ is the common ratio.

Strictly Increasing Sequence

A sequence $\{a_n\}$ satisfying the strict inequality $a_{n+1} > a_n$ for all $n \ge 1$.

Strictly Decreasing Sequence

A sequence $\{a_n\}$ satisfying the strict inequality $a_{n+1} < a_n$ for all $n \ge 1$.

Constant Sequence

A sequence that is simultaneously increasing and decreasing, meaning there exists a constant $k \in \mathbb{R}$ such that $a_n = k$ for all $n \ge 1$.

Bounded Sequence

A sequence $\{a_n\}$ that is bounded both from above and from below, meaning there exists a real number $k > 0$ such that $|a_n| \le k$ for all $n \ge 1$.

Eventually Satisfied Property

A condition or property $P$ that holds for all terms of a sequence starting from a specific index $\bar{n} = \bar{n}_P$, such that the property is satisfied for all $n \ge \bar{n}$.

Convergent Sequence

A sequence $\{a_n\}$ that approaches a real number $\lambda \in \mathbb{R}$ such that for every $\epsilon > 0$, there exists an integer $n_\epsilon \ge 1$ with $|a_n - \lambda| < \epsilon$ for all $n \ge n_\epsilon$.

Infinitesimal (Null) Sequence

A sequence $\{a_n\}$ whose terms converge to zero, meaning $\lim_{n \rightarrow +\infty} a_n = 0$.

Positively Divergent Sequence

A sequence $\{a_n\}$ that eventually exceeds any real threshold, meaning for every $k \in \mathbb{R}$ there exists $n_k \ge 1$ such that $a_n > k$ for all $n \ge n_k$, denoted $a_n \rightarrow +\infty$.

Negatively Divergent Sequence

A sequence $\{a_n\}$ that eventually falls below any real threshold, meaning for every $k \in \mathbb{R}$ there exists $n_k \ge 1$ such that $a_n < k$ for all $n \ge n_k$, denoted $a_n \rightarrow -\infty$.

Irregular (Oscillating) Sequence

A sequence that does not possess any limit in the extended real line $\overline{\mathbb{R}}$, neither converging to a finite value $\lambda \in \mathbb{R}$ nor diverging to $+\infty$ or $-\infty$.

Limit from Above

The limit behavior denoted $\lim_{n \rightarrow +\infty} a_n = \lambda^+$ or $a_n \rightarrow \lambda^+$, indicating that $a_n \rightarrow \lambda$ while remaining greater than or equal to $\lambda$ ($a_n \ge \lambda$) eventually.

Limit from Below

The limit behavior denoted $\lim_{n \rightarrow +\infty} a_n = \lambda^-$ or $a_n \rightarrow \lambda^-$, indicating that $a_n \rightarrow \lambda$ while remaining less than or equal to $\lambda$ ($a_n \le \lambda$) eventually.

Uniqueness of the Limit Theorem

A fundamental theorem stating that a sequence $\{a_n\}$ can converge to at most one limit $\lambda \in \overline{\mathbb{R}}$.

Comparison Criterion (Squeeze Theorem)

A theorem stating that if three sequences satisfy $a_n \le b_n \le c_n$ eventually and $\lim_{n \rightarrow +\infty} a_n = \lim_{n \rightarrow +\infty} c_n = \lambda \in \overline{\mathbb{R}}$, then $\lim_{n \rightarrow +\infty} b_n = \lambda$.

Ratio Criterion

A convergence test stating that if there exists a scalar $q < 1$ such that eventually $|\frac{a_{n+1}}{a_n}| \le q$, then $\lim_{n \rightarrow +\infty} a_n = 0$.

Regularity Theorem for Monotonic Sequences

A theorem stating that every monotonic sequence is regular: it converges if it is bounded, diverges to $+\infty$ if increasing and unbounded, and diverges to $-\infty$ if decreasing and unbounded.

Indeterminate Forms

The seven undefined algebraic expressions encountered in limit computations: $0 \cdot (\pm\infty)$, $\infty - \infty$, $\frac{0}{0}$, $\frac{\pm\infty}{\pm\infty}$, $0^0$, $\infty^0$, and $1^{\pm\infty}$.

Little-o Notation

The notation $b_n = o(a_n)$, meaning that $\lim_{n \rightarrow +\infty} \frac{b_n}{a_n} = 0$, which defines sequence $\{b_n\}$ as negligible with respect to sequence $\{a_n\}$.

Asymptotic Sequences

Two sequences $\{a_n\}$ and $\{b_n\}$ that satisfy $\lim_{n \rightarrow +\infty} \frac{b_n}{a_n} = 1$, denoted by $b_n \sim a_n$, which holds if and only if $a_n = b_n + o(b_n)$.

Infinity of Higher Order

A divergent sequence $a_n \rightarrow +\infty$ relative to another infinity $b_n \rightarrow +\infty$ satisfying $\lim_{n \rightarrow +\infty} \frac{a_n}{b_n} = +\infty$, demonstrating that $a_n$ diverges to $+\infty$ faster than $b_n$.

Hierarchy (Scale) of Infinities

The established growth ranking among infinities for large $n$: $\ln^k(n) = o(n^\alpha)$ for powers ($\alpha > 0$), $n^k = o(a^n)$ for exponentials ($a > 1$), $a^n = o(n!)$, and $n! = o(n^n)$.

Infinitesimal of Higher Order

An infinitesimal sequence $a_n \rightarrow 0$ relative to another infinitesimal $b_n \rightarrow 0$ such that $\lim_{n \rightarrow +\infty} \frac{a_n}{b_n} = 0$, indicating that $a_n$ tends to zero faster than $b_n$.

Napier's Constant ($e$)

The mathematical constant defined by the limit $e = \lim_{n \rightarrow +\infty} (1 + \frac{1}{n})^n$, which has an approximate value of $2.71828\dots$.

Fundamental Limit for $(1 + \frac{a}{n})^n$

The general limit identity stating that for any real scalar $a \in \mathbb{R}$, $\lim_{n \rightarrow +\infty} (1 + \frac{a}{n})^n = e^a$.

Convergence of Vector Sequences

A sequence of vectors $\{a^k\}$ in $\mathbb{R}^n$ converges to $\lambda \in \mathbb{R}^n$ if and only if the scalar sequence of Euclidean distances satisfies $\lim_{k \rightarrow +\infty} \|a^k - \lambda\| = 0$, which is equivalent to every component sequence converging to the corresponding component $\lambda_i$.