전체 글

Theorem 24.1Theorem 24.1. Let $\sum_{n=1}^{\infty} a_n$ be a series with nonnegative terms. Then $\sum_{n=1}^{\infty} a_n$ converges $\iff$ the sequence of partial sums $\{ s_n \}$ is bounded. Proof. $(\Longrightarrow)$ Since $\sum_{n=1}^{\infty} a_n = \lim_{n \to \infty} s_n$ converges, $\{ s_n \}$ is bounded by Theorem 13.2.$(\Longleftarrow)$Since $a_n \geq 0, \forall n \in \mathbb{P}$, $\{ s_..

Infinite SeriesDefinition 22.1. Let $\{a_n\}$ be a sequence. For each positive integer $n$, let $$s_n = a_1 + a_2 + \cdots + a_n = \sum_{k=1}^{n} a_k.$$ An infinite series is the ordered pair of sequences $(\{a_n\}, \{s_n\})$.$s_n$ is called the $n$th partial sum of the infinite series. Convergent SeriesDefinition 22.2. Let $\sum_{n=1}^{\infty} a_n$ be an infinite series. If the sequence of pa..

NoteNote. We will extend our definition of lim sup and lim inf to the unbounded sequences.Let $\{ a_n \}$ be a bounded sequence. Then $$A_n = \sup \{ a_n, a_{n+1}, ... \}$$ exists for every positive integers $n$. Since $\{ a_{n+1}, a_{n+2}, ... \} \subset \{ a_n, a_{n+1}, ... \}$, $A_n \geq A_{n+1}$ for all positive integers $n$, which means that $\{ A_n \}$ is decreasing. Since $\{ a_n \}$ is b..

Definition 20.1Definition 20.1. Let $\{a_n\}$ be a bounded real sequence and let $\mathcal{L}_a$ denote the set of all $L$ such that $$L = \lim_{k \to \infty} a_{n_k}$$ where $\{a_{n_k}\}$ is a convergent subsequence of $\{a_n\}$. We define $$\limsup_{n \to \infty} a_n = \sup \mathcal{L}_a$$ and $$\liminf_{n \to \infty} a_n = \inf \mathcal{L}_a$$ The notations \(\overline{\lim}_{..

Theorem 19.1Theorem 19.1. Let $\{a_n\}$ be a convergent sequence. Then for every $\varepsilon > 0$, there exists a positive integer $N$ such that if $m, n \geq N$, then \[ |a_m - a_n| Proof. Let $\{ a_n \}$ be a sequence with the limit $L$ and let $\varepsilon > 0$. Then $\exists N \in \mathbb{P}$ such that $|a_n - L| Definition 19.2Definition 19.2. If $\{a_n\}$ is a sequence such that..

The Bolzano-Weierstrass TheoremTheorem 18.1 (The Bolzano-Weierstrass Theorem). Every bounded real sequence has a convergent subsequence. Proof. Let $\{ a_n \}$ be a bounded real sequence. Then there exists a closed interval $[ c, d ]$ such that $a_n \in [ c, d ], \forall n \in \mathbb{P}$. Consider the two subinterval, $\left[ c, \frac{c+d}{2} \right], \left[ \frac{c+d}{2}, d \right]$. One of th..

Theorem 17.1 Theorem 17.1. If $x$ is a real number, there exists an increasing rational sequence $\{ r_n \}$ with limit $x$. Proof. By Theorem 7.8, $\exists r_1 \in \mathbb{Q}$ such that $x-1 Note that$$x - \frac{1}{n} RemarkRemark. If$a \geq 1$and$x$is a real number, we choose an increasing rational sequence$\{ r_n \}$such that$\lim_{n \to \infty} r_n = x$. Since$r_n \leq r_{n+1}$,$a^..

Monotone SequencesDefinition 16.1. Let $\{a_n\}$ be a sequence. We say that $\{a_n\}$ is increasing (decreasing) if $a_n \leq a_{n+1}$ ($a_n \geq a_{n+1}$) for every positive integer $n$. We say that the sequence $\{a_n\}$ is monotone if either $\{a_n\}$ is increasing or $\{a_n\}$ is decreasing. If $a_n a_{n+1}$) for every positive integer $n$, we say that $\{a_n\}$ is strictly increasing (stri..

Divergent SequencesDefinition 15.1. Let $\{a_n\}$ be a sequence. We say that $\{a_n\}$ diverges to infinity (or minus infinity) and write $$\lim_{n \to \infty} a_n = \infty \quad (\lim_{n \to \infty} a_n = -\infty)$$ if for every real number $M$, there exists a positive integer $N$ such that if $n \geq N$, then $a_n > M$ ($a_n Theorem 15.2Theorem 15.2. Let$\{a_n\}$and$\{b_n\}$ be sequences ..

Bounded Sequences Definition 13.1. We say that a sequence $\{a_n\}$ is bounded above (below) if there exists a number $M$ such that $a_n ≤ M$ ($a_n \geq M$) for every positive integer $n$. We say that $\{a_n\}$ is bounded if $\{a_n\}$ is bounded both above and below. Remark. A sequence $\{a_n\}$ is bounded $\iff$ $\exists M > 0$ such that $|a_n| \leq M, \forall n \in \mathbb{P}$. Theorem 13.2 T..