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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 arithmetic 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.

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

\[ |x_n - x| < \varepsilon \]

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 \((x_n)\) and \((y_n)\) 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 \(|x_n| \rightarrow |x|\)
  2. If \(|x_n| \rightarrow 0\), then \(x_n \rightarrow 0\)
  3. \(x_n \rightarrow x\) if and only if \(|x_n - n| \rightarrow 0\)

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 \(\varepsilon\), there exists a natural number \(H\) such that all natural numbers \(m, n \geq H\), then

\[|x_m - x_n \leq \varepsilon\]

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 generated 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.

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 \(\varepsilon\), there exists some natural number \(H\) such that when \(m > n > H\),

\[ |s_m - s_n| = |\sum_{i = m + 1}^n x_i| < \varepsilon \]

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

Corollary. Absolute Convergence Test. If \(\sum |x_n|\) converges, then \(\sum x_n\) converges.

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

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