전체 글

Taylor Series함수 $f$를 양수의 수렴 반경을 가지고 $$f(x) = \sum_{n=0}^{\infty} a_n(x-a)^n$$ 이라고 하자. 이 수렴 구간에서 미분하면 $$f'(x) = a_1 + 2a_2(x-a) + 3a_3(x-a)^2 + \cdots + na_n(x-a)^{n-1} + \cdots , \\ f''(x) = 1\cdot 2 a_2 + 2 \cdot 3 a_3 (x-a) + 3\cdot 4 a_4 (x-a)^2 + \cdots , \\ f'''(x) = 1 \cdot 2 \cdot 3 a_3 + 2 \cdot 3 \cdot 4a_4(x-a) + 3\cdot 4 \cdot 5a_5 (x-a)^2 + \cdots , \\ \vdots$$ 임을 알 수 있다. 각 등식에 $x=a$..

Power SeriesDefinition 1. A power series about $x=a$ is a series of the form $$\sum_{n=0}^{\infty} c_n(x-a)^n$$ in which the center $a$ and the coefficients $c_0, c_1, ..., c_n$ are constants.Convergent Theorem for Power SeriesTheorem 1. If the power series $\sum_{n=0}^{\infty} a_n x^n$ converges at $x = c \neq 0$, then it converges absolutely for all $x$ with $|x| |d|$.Corollary. The convergen..

The $n$th-Term Test for a Divergent Series Theorem 1. If $\sum_{n=1}^{\infty} a_n$ converges, then $a_n \rightarrow 0$.Proof. Let $\sum_{n=1}^{\infty} a_n = \lim_{n \to \infty} s_n = L$, where $s_n$ is the partial sums of the series and $L$ is the sum of the series. Note that $$\lim_{n \to \infty} s_n = \lim_{n \to \infty} s_{n-1} = L \\ \Longrightarrow \lim_{n \to \infty} (s_n - s_{n-1}) = \lim..

Ordered SetDefinition 1. An order $relation with the following two properties:(1) If$x \in S$and$y \in S$, then one and only one of the statements$$x(2)$S$is transitive.We call$S$an ordered set if an order is defined in$S$.BoundedDefinition 2. Suppose$S$is an ordered set, and$E \subset S$. (1) If there exists a$\beta \in S$such that$x \leq \beta, \forall x \in E$, we say that$E$ ..

Equivalence RelationDefinition 1. Let $R$ be a relation in a set $X$. Then we say that (a) $R$ is reflexive $\iff$ $\forall x \in X, xRx$.(b) $R$ is symmetric $\iff$ $xRy \Longrightarrow yRx$.(c) $R$ is transitive $\iff$ $xRy \wedge yRz \Longrightarrow xRz$. (d) $R$ is an equivalence relation $\iff$ $R$ is reflexive, symmetric, and transitive. Equivalence relation, 즉 동치 관계는 특정한 수학적 관점에서 볼 때 두 원소..

Partial OrderDefinition 1. A relation $\leq$ on a set $A$ is called a partial order relation if and only if the relation $\leq$ is reflexive and transitive on $A$ and antisymmetric on $A$, that is, if $a\leq b$ and $b \leq a$, then $a = b$. A partially ordered set is a pair $(A, \leq)$, where $A$ is a set and $\leq$ is a partial order relation on $A$.Total orderDefinition 2. A total order relati..

RelationDefinition 1. A relation $R$ from $A$ to $B$ is a subset of $A \times B$. It is customary to write $aRb$ for $(a, b) \in R$. The symbol $aRb$ is read $a$ is $R$-related to $b$.많은 경우 $A = B$이며, 이때 관계 $R$은 relation in $A$ 라고 말한다. Inverse RelationDefinition 2. Let $A, B$ be sets, not necessarily distinct, and let $R$ be a relation from $A$ to $B$. Then inverse $R^{-1}$ of $R$ is the relatio..

SequenceDefinition 1. A sequence is a function whose domain is $\mathbb{N}$.고등학교에서는 수열을 '수의 나열'이라고 정의하곤 하는데, 정의에 의하면 꼭 '수'를 나열한 것만이 수열이 될 필요는 없다. 수가 아닌 함수나 다른 대상도 가능하다. Bounded SequenceDefinition 2. A sequence $\{ a_n \}$ is said to be bounded if its range is bounded. That is, there exists a number $M > 0$ such that $|a_n| \geq M$ for all $n \in \mathbb{N}$.Monotonic SequenceDefinition 3. A sequ..

Infinite SeriesDefinition 1. Given a sequence of numbers $\{ a_n \}$, an expression of the form $$a_1 + a_2 + \cdots + a_n + \cdots$$ is an infinite series. The number $a_n$ is the $n$th term of the series. The sequence $\{ s_n \}$ defined by $$s_n = \sum_{k=1}^n a_k$$ is the sequence of partial sums of the series, the number $s_n$ being the $n$th partial sum. If the sequence of partial sums con..

Sequence, 즉 수열은 숫자들의 나열이라고 정의할 수 있고 $$\{a_n\}_{n=1}^\infty$$ 로 표기되는 무한수열은 정의역이 자연수인 함수로 간주할 수 있다.Convergence and Divergence of SequencesDefinition 1. The sequence $\{ a_n \}$ converges to the number $L$ if for every positive number $\epsilon$ there corresponds an integer $N$ such that for all $n$, $$n > N \Longrightarrow |a_n - L| diverges. If $\{ a_n \}$ converges to $L$, we write $\lim_{n \righ..