notes-archive/docs/math/real-analysis/3-sequences-series.md

Chapter 3 - Sequences and Series

Section 3.1 - Sequences and their Limits

Definition. A sequence in $\mathbb{R}$ is a function X: \mathbb{N} \rightarrow \mathbb{R}, typically notated as X or (x_n), with x_n being referred to as the terms of the sequence. The set {x_n | n \in \mathbb{N}} is the range of this sequence.

Definition. The sequence is bounded if its range is a bounded subset of \mathbb{R}.

Example. The constant sequence C = (c) = (c, c, c, \ldots).

Example. The harmonic sequence \frac{1}{n} = (1, \frac{1}{2}, \frac{1}{3}, \ldots)

Example. The geometric sequence, given base a \in \mathbb{R} and ratio r \in \mathbb{R}

(x_n) = (a, ar,  ar^2, ar^3, \ldots)

Example. The arithmatic sequence, given base a \in \mathbb{R} and distance d \in \mathbb{R},

(x_n) = (a, a + d, a + 2d, a + 3d, \ldots)

Example. Decimal expansions are bounded sequences.

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Definition. A sequence X = (x_n) is said to converge to a number x \in \mathbb{R} if when given any \epsilon > 0, there exists some K \in \mathbb{N} such that for every n \in \mathbb{N} with n \geq K,

\abs{x_n - x} < \epsilon

If this is the case, we say that X converges to x, and x is a limit of X. This can be written as

\lim X = x \text{ or } \text \lim(x_n) = x \text{ or }  x_n \rightarrow x

Definition. If a sequence does not have a limit, it is divergent.

Theorem. A sequence can have at most one limit. That is, if a limit exists, it is unique.

Theorem. If a limit is convergent, then it is bounded.

Section 3.2 - Limit Theorems

Theorem. Suppose there exists some X such that (x_n) \rightarrow x and Y such that (y_n) \rightarrow y. Then,

1. x_n + y_n \rightarrow x + y
2. x_n \cdot y_n \rightarrow xy
3. If x_n \neq 0 for all n, then \frac{1}{x_n} \rightarrow \frac{1}{x}

Theorem. Suppose (x_n) aand (y_n) are convergent sequences and N \in \mathbb{N}. Then,

1. If x_n \leq y_n for all n \geq N, then \lim(x_n) \leq \lim(y_n)
2. If x_n \leq a for all n \geq N, then \lim(x_n) \leq a
3. If x_n \geq a for all n \geq N, then \lim(x_n) \geq a

Theorem. Squeeze Theorem. Suppose (x_n), (y_n), (z_n) are all sequences of real numbers, and \lim(x_n) = \lim(z_n) = a. Then, if for some N in \mathbb{N},

\text{If } x_n \leq y_n \leq z_n, \text{ then } \lim(y_n) = a

Theorem. Suppose (x_n) is a sequence if real numbers. Then,

1. If x_n \rightarrow x, then \abs{x_n} \rightarrow \abs{x}
2. If \abs{x_n} \rightarrow 0, then x_n \rightarrow 0
3. x_n \rightarrow x if and only if \abs{x_n - n} \rightarrow 0

Theorem. Suppose (x_n) is a sequence if real numbers, with each x_n \geq 0. Then, given some k \in \mathbb{N}, if x_n \rightarrow x, then \sqrt[k]{x_n} \rightarrow \sqrt[k]{x}.

Section 3.3 - Monotonic Sequences

Definition. A sequence (x_n) is monotonically increasing if x_{n+1} \geq x_n for all n \in \mathbb{N}.

Definition. A sequence (x_n) is monotonically decreasing if x_{n+1} \leq x_n for all n \in \mathbb{N}.

Definition. A sequence is monotonic if it is either monotonically increasing or decreasing.

Theorem. A monotonic sequence is converging if and only if it is bound.

Section 3.4 - Subsequences

Definition. Let X = (x_n) be a sequence in \mathbb{R}. Then, the sequence

X_{n_k} = (x_{n_1}, x_{n_2}, \ldots)

is a subsequence of X,

Theorem. If a sequence converges to x, then every subsequence also converges to x.

Theorem. Every sequence of real numbers (x_n) contains a monotonic subsequence (x_{n_k}).

Corollary. Bolzano-Weierstrass Theorem. Every bounded sequence of real numbers has a convergent subsequence.

Section 3.5 - The Cauchy Criterion

Definition. A sequence (x_n) is said to be a Cauchy sequence such that for any given \epsilon, there exists a natural number H such that all natural numbers m, n \geq H, then

\abs{x_m - x_n} \leq \epsilon

Theorem. If (x_n) is a Cauchy sequence, then (x_n) is convergent.

Section 3.7 - Series

Definition. Let (x_n) be a sequence in \mathbb{R}. Then, the infinite series genearted by $X$ is the sequence S = (s_n) with terms

s_1 = x_1; \; s_{n+1} = s_n + x_{n+1}

In other words, s_n = \sum_{i=1}^n x_i. We denote this series as \sum x_n.

Definition. If this series is convergent to some number s, we say that s is the sum of the series.

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For natural numbers n > m, note that

s_n - s_m = \sum_{i=m + 1}^n x_i

In particular, s_n - s_{n - 1} = x^n. Thus, the Cauchy criteria takes the form

Theorem. Cauchy Criteria for Series. The series \sum x_n converges if and only if, for a given \epsilon, there exists some natural number H such that when m > n > H,

\abs{s_m - s_n} = \abs{\sum_{i = m + 1}^n x_i} < \epsilon

Corollary. $n$-th Term Test. If \sum x_n converges, then x_n \rightarrow 0.

Corollary. Absolute Convergence Test. If \sum \abs{x_n} converges, then \sum x_n converges.

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Theorem. A series with non-negative terms converges if and only if its sequence of partial sums is bounded.

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Theorem. e = \lim_{n \rightarrow \infty} (1+\frac{1}{n})^n = \sum_{n=0}^\infty \frac{1}{n!}