Accumulation PointDefinition 30.1. Let $X \subset \mathbb{R}$ and let \( a \in \mathbb{R} \). We say that \( a \) is an accumulation point of \( X \) if for every \( \delta > 0 \), there exists a number \( x \in X \) such that \( 0 We say that $a$ is a left (right) accumulation point of $X$ if for every $\delta > 0$, there exists a number $x \in X$ such that $0 다른말로, $a$ 근방에 $a$와는 다른 $x \in X$가 ..

Theorem 28.1Theorem 28.1 (Summation by Parts). Let \( \sum_{n=1}^{\infty} a_n \) be an infinite series and let \( \{s_n\} \) be the sequence of partial sums of \( \sum_{n=1}^{\infty} a_n \). Let \( \{b_n\} \) be any sequence. Then for any positive integer \( n \), we have \[ \sum_{k=1}^{n} a_k b_k = \sum_{k=1}^{n} s_k (b_k - b_{k+1}) + s_n b_{n+1}. \]Proof. Let $s_0 = 0$. Then $$\sum_{k=1}^n a_k..

Power SeriesDefinition 27.1. Let \( t \) be a fixed real number. A power series (expanded about \( t \)) is an infinite series of the form \[ \sum_{n=0}^{\infty} a_n (x - t)^n \] where \( \{a_n\}_{n=0}^{\infty} \) is a sequence and \( x \) is a real number. \([(x - t)^0\) is defined to be 1.] Theorem 27.2Theorem 27.2. Let \( \sum_{n=0}^{\infty} a_n (x - t)^n \) be a power series. Let \[ L = \lim..

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^..