Limits and Convergence¶

Sequences¶

A sequence of real numbers is a mapping $\mathbb{N} \to \mathbb{R}$, written as $(a_n)_{n \geq 1}$ or simply $(a_n)$. The number $a_n$ is called the $n$-th term of the sequence.

Example. The sequence $a_n = \frac{1}{n}$ gives $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots$

Convergence¶

A sequence $(a_n)$ converges to the limit $L \in \mathbb{R}$ if for every $\varepsilon > 0$ there exists an $N \in \mathbb{N}$ such that:

\[ |a_n - L| < \varepsilon \quad \text{for all } n \geq N \]

Notation: $\lim_{n \to \infty} a_n = L$ or $a_n \to L$.

The terms of the sequence approach $L$ to arbitrary precision — from some index onward, all terms lie in the interval $(L - \varepsilon, L + \varepsilon)$.

Example. $\lim_{n \to \infty} \frac{1}{n} = 0$, since for every $\varepsilon > 0$, $\frac{1}{n} < \varepsilon$ whenever $n > \frac{1}{\varepsilon}$.

A sequence that does not converge is called divergent.

Rules of Computation¶

For convergent sequences with $a_n \to A$ and $b_n \to B$:

$a_n + b_n \to A + B$
$a_n \cdot b_n \to A \cdot B$
$\frac{a_n}{b_n} \to \frac{A}{B}$ (provided $B \neq 0$)

Series¶

An (infinite) series is the sequence of partial sums of a sequence $(a_k)$:

\[ S_n = \sum_{k=1}^{n} a_k = a_1 + a_2 + \cdots + a_n \]

The series $\sum_{k=1}^{\infty} a_k$ converges if the sequence $(S_n)$ converges:

\[ \sum_{k=1}^{\infty} a_k = \lim_{n \to \infty} S_n \]

Geometric Series¶

The most important series in elementary analysis: for $|q| < 1$:

\[ \sum_{k=0}^{\infty} q^k = \frac{1}{1-q} \]

Example. $\sum_{k=0}^{\infty} \left(\frac{1}{2}\right)^k = \frac{1}{1 - 1/2} = 2$.

For $|q| \geq 1$, the geometric series diverges.

Necessary Condition¶

If $\sum a_k$ converges, then $a_k \to 0$. The converse is false: the harmonic series $\sum_{k=1}^{\infty} \frac{1}{k}$ diverges, even though $\frac{1}{k} \to 0$.

Cauchy Sequences¶

A sequence $(a_n)$ is a Cauchy sequence if its terms become arbitrarily close to each other: for every $\varepsilon > 0$ there exists an $N$ such that:

\[ |a_m - a_n| < \varepsilon \quad \text{for all } m, n \geq N \]

"The concept of Cauchy sequence captures convergence without reference to the limit itself." — Walter Rudin, Principles of Mathematical Analysis, McGraw-Hill, 1976.

In $\mathbb{R}$: a sequence converges if and only if it is a Cauchy sequence. This property is called the completeness of $\mathbb{R}$.

Completion¶

Not every metric space is complete. The completion of a space constructs a larger space in which all Cauchy sequences converge.

The central example in number theory: the rational numbers $\mathbb{Q}$ are not complete. Their completion with respect to the ordinary absolute value yields $\mathbb{R}$. Their completion with respect to the $p$-adic absolute value yields the $p$-adic numbers $\mathbb{Q}_p$ — an entirely different number field that plays a central role in modern number theory.

Summary¶

Concept	Definition
Convergence	$\lim_{n\to\infty} a_n = L$: $\forall\varepsilon>0\ \exists N: n\geq N \Rightarrow
Series	$\sum_{k=1}^{\infty} a_k = \lim_{n\to\infty} \sum_{k=1}^{n} a_k$
Geometric series	$\sum_{k=0}^{\infty} q^k = \frac{1}{1-q}$ for $
Cauchy sequence	$\forall\varepsilon>0\ \exists N: m,n\geq N \Rightarrow
Completeness	Every Cauchy sequence converges
Completion	Construction of a complete space from an incomplete one

References¶

Rudin, Walter: Principles of Mathematical Analysis. McGraw-Hill, 3rd edition, 1976. Chapter 3.
Bartle, Robert G.; Sherbert, Donald R.: Introduction to Real Analysis. Wiley, 4th edition, 2011. Chapters 3–4.

Concept	Definition
Convergence	\(\lim_{n\to\infty} a_n = L\): $\forall\varepsilon>0\ \exists N: n\geq N \Rightarrow
Series	\(\sum_{k=1}^{\infty} a_k = \lim_{n\to\infty} \sum_{k=1}^{n} a_k\)
Geometric series	\(\sum_{k=0}^{\infty} q^k = \frac{1}{1-q}\) for $
Cauchy sequence	$\forall\varepsilon>0\ \exists N: m,n\geq N \Rightarrow
Completeness	Every Cauchy sequence converges
Completion	Construction of a complete space from an incomplete one