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..

Improper Integrals Definition 1. Integrals with infinite limits of integration are improper integrals of Type I.(1) If $f(x)$ is continuous on $[0, \infty)$, then $$\int_a^{\infty} f(x) dx = \lim_{b \rightarrow \infty} \int_a^b f(x) dx.$$ (2) If $f(x)$ is continuous on $(- \infty, b]$, then $$\int_{- \infty}^b f(x) dx = \lim_{a \rightarrow - \infty} \int_a^b f(x) dx.$$ (3) If $f(x)$ is continuou..

IrreducibleDefinition 1. A quadratic polynomial is irreducible if it cannot be written as the product of two linear factors with real coefficients. That is, the polynomial has no real roots. The Fundament든 Theorem of Algebra, 대수학의 기본정리에 의해 모든 실계수 다항식은 irreducible polynomial, 즉 linear or quadratic polynomial로 분해될 수 있다는 사실이 증명되어 있다. Method of Partial FractionsFor polynomials $f(x)$ and $g(x)$ with..

intervalreal root

Integration by PartsTheorem 1. Let $f$ and $g$ be differentiable functions of $x$. Then $$\int f(x) g'(x) dx = f(x) g(x) - \int f'(x) g(x) dx.$$Proof. By the Product Rule, we have $$\frac{d}{dx} [f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x) \\ \Longrightarrow \int \frac{d}{dx} [f(x) \cdot g(x)] = \int f'(x) g(x) dx + \int f(x) g'(x) dx \\ \Longrightarrow \int f(x) g'(x) = f(x) g(x) - \int f'(x) g(x)..

Variable DeclarationLet $x$ be a~Let $F$ denote any~For a real number $x$, ~Supposing PartLet suppose that ~Let assume that ~Suppose that ~Assume that ~Definition PartWe define ~ by ~, as~We say that ~A ~~ is said to be ~~ is referred to as ~

LinearizationDefinition 1. If $f$ is differentiable at $x=a$, then the approximating function $$L(x) = f(a) + f'(a)(x-a)$$ is the linearization of $f$ at $a$. The approximation $$f(x) \approx L(x)$$ of $f$ by $L$ is the standard linear approximation of $f$ at $a$. The point $x=a$ is the center of the approximation.미분가능한 함수 $f$에 대해 $(a, f(a))$에 접하는 접선의 방정식은 위와 같이 주어지며, 이를 선형 근사라고 부른다. Differentia..

Indeterminate Forms종종 극한을 계산할 때 $\frac{0}{0}$, 혹은 $\frac{\infty}{\infty}$, $\infty \cdot 0, \infty - \infty, 0^0, 1^{\infty}$와 같은 꼴로 나타날 때가 있다. 이러한 형태의 극한은 명백한 어느 한 값으로 수렴한다고 단정 짓기 어려우며, 따라서 이런 형태들을 indeterminate forms, 즉 부정형이라고 한다. L'Hospital's Rule로피탈의 정리는 이러한 부정형을 계산하는 방법 중 하나이다. Theorem 1. Suppose that $f(a) = g(a) = 0$, that $f$ and $g$ are differentiable on an open interval $I$ containing ..