Because of the introduction of zero, it became possible to represent rational numbers in decimal forms as in \(\frac{22}{7} = 3.142857 \cdots\) . We use this as a motivation for studying sequences to better understand the dots in “ \(3.142857 \cdots\) ”. We give a formal definition of sequences in Sect. 3.1, and explain why the sequence notation \(\left\langle {a_{n} } \right\rangle_{n = 1}^{\infty }\) should not be confused with the set notation \(\left\{ {a_{n} } \right\}_{n = 1}^{\infty }\) . We then define the convergence of a sequence, give examples using this definition, and prove some basic theorems. In order to test convergence of sequences, we introduce only a few but very fundamental theorems in Sect. 3.2. In Sect. 3.3, series are introduced as a method of obtaining a new sequence from an existing sequence. We do not distinguish series from sequences, and therefore, we emphasize the theorems introduced in Sects. 3.1 and 3.2, and we apply them to series of positive terms in Sect. 3.3. In Sects. 3.4 and 3.5, we introduce series of mixed (positive or negative) terms. In order to better understand real numbers, we introduce the binary and ternary number systems in Sect. 3.6. There, we practice additions, subtractions, multiplications, and divisions of numbers in bases two and three. This leads us to the introduction of the Cantor set and Cantor functions. We prove Bolzano-Weierstrass Theorem and Cauchy sequences are introduced in Sect. 3.7. In Sect. 3.8, we will prove the existence of exponential numbers of the type \(\sqrt[n]{x}\) as a generalization of the existence proof of \(\sqrt 2\) in Chap. 2 . This enables us to define \(x^{r}\) when \(r\) is a rational number. In Sect. 3.9, we define \(x^{r}\) when \(r\) is an irrational number as an application of Cauchy sequences. In Sect. 3.10, we define the number \(e\) to be \(\mathop \sum \limits_{n = 0}^{\infty } \frac{1}{n!}\) , and prove that \(e = \mathop {\lim }\limits_{m \to \infty } \left( {1 + \frac{1}{m}} \right)^{m}\) , which is a more common definition of \(e\) . In Sect. 3.11, we offer a way to generalize what we have studied so far and apply it to metric spaces.

Sequences and Series

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发表日期：

2025年

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Hidefumi Katsuura

摘要

Because of the introduction of zero, it became possible to represent rational numbers in decimal forms as in \(\frac{22}{7} = 3.142857 \cdots\) . We use this as a motivation for studying sequences to better understand the dots in “ \(3.142857 \cdots\) ”. We give a formal definition of sequences in Sect. 3.1, and explain why the sequence notation \(\left\langle {a_{n} } \right\rangle_{n = 1}^{\infty }\) should not be confused with the set notation \(\left\{ {a_{n} } \right\}_{n = 1}^{\infty }\) . We then define the convergence of a sequence, give examples using this definition, and prove some basic theorems. In order to test convergence of sequences, we introduce only a few but very fundamental theorems in Sect. 3.2. In Sect. 3.3, series are introduced as a method of obtaining a new sequence from an existing sequence. We do not distinguish series from sequences, and therefore, we emphasize the theorems introduced in Sects. 3.1 and 3.2, and we apply them to series of positive terms in Sect. 3.3. In Sects. 3.4 and 3.5, we introduce series of mixed (positive or negative) terms. In order to better understand real numbers, we introduce the binary and ternary number systems in Sect. 3.6. There, we practice additions, subtractions, multiplications, and divisions of numbers in bases two and three. This leads us to the introduction of the Cantor set and Cantor functions. We prove Bolzano-Weierstrass Theorem and Cauchy sequences are introduced in Sect. 3.7. In Sect. 3.8, we will prove the existence of exponential numbers of the type \(\sqrt[n]{x}\) as a generalization of the existence proof of \(\sqrt 2\) in Chap. 2 . This enables us to define \(x^{r}\) when \(r\) is a rational number. In Sect. 3.9, we define \(x^{r}\) when \(r\) is an irrational number as an application of Cauchy sequences. In Sect. 3.10, we define the number \(e\) to be \(\mathop \sum \limits_{n = 0}^{\infty } \frac{1}{n!}\) , and prove that \(e = \mathop {\lim }\limits_{m \to \infty } \left( {1 + \frac{1}{m}} \right)^{m}\) , which is a more common definition of \(e\) . In Sect. 3.11, we offer a way to generalize what we have studied so far and apply it to metric spaces.