Historical Background of Infinite Series

The Problem: Area of a Circle

Imagine being a 17th-century mathematician facing a simple-looking question: How can we compute the area of a circle exactly?

The formula is $A = \pi r^2$, but what is $\pi$ precisely? People knew $\pi \approx 3.14$ and needed more digits.

Limits of Classical Geometry

Archimedes estimated $\pi$ by inscribed and circumscribed polygons (hexagon, dodecagon, etc.), but this was laborious and only modestly accurate.

The Geometric-Series Formula

Mathematicians noticed

$\frac{1}{1 - x} = 1 + x + x^2 + x^3 + x^4 + \cdots$

When $x = \frac{1}{2}$,

$\frac{1}{1 - \frac{1}{2}} = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots = 2,$ an infinite addition giving a finite result.

If you want Euler’s proof:

Euler's proof idea

1. Let $S = 1 + x + x^2 + x^3 + \cdots$
2. Multiply by $x$: $xS = x + x^2 + x^3 + \cdots$
3. Subtract: $S - xS = 1$
4. Factor: $(1 - x)S = 1$
5. Solve: $S = \frac{1}{1 - x}$

Converges when $|x| < 1$. The algebra is simple; its insight was profound for later analysis.

Birth of Infinite Series

This showed a finite number can be expressed as an infinite sum—opening the door to infinite series.

Computing $\pi$

Leibniz later found

$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots,$ derived from the power series of $\arctan x$ at $x=1$—simple operations to approximate $\pi$, though slowly convergent.

Leibniz discovery sketch

1. Use $\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots$
2. Substitute $x = 1$ to get $\arctan 1 = \frac{\pi}{4}$
3. Hence $\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots$

Historically the first infinite-series formula for $\pi$.

练习题

练习 1

Compute the first four-term sum of $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8}$.

参考答案

Sum: $1 + 0.5 + 0.25 + 0.125 = 1.875$.

练习 2

Use the Leibniz series to approximate $\pi$ with the first three terms.

参考答案

$\frac{\pi}{4} \approx 1 - \frac{1}{3} + \frac{1}{5} = 0.8667$
$\pi \approx 4 \times 0.8667 \approx 3.4668$.

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总结

本文出现的符号

符号	类型	读音/说明	在本文中的含义
$\pi$	希腊字母	Pi（派）	圆周率，约等于 3.14159
$\sum$	希腊字母	Sigma（西格玛）	求和符号，表示级数
$\infty$	数学符号	无穷大	表示无穷级数，项数无限
$x$	数学符号	变量	级数中的变量

中英对照

中文术语	英文术语	音标	说明
圆周率	pi	/paɪ/	圆的周长与直径的比值，记作 $\pi$
几何级数	geometric series	/dʒiːəˈmetrɪk ˈsɪəriːz/	形如 $\sum_{n=0}^{\infty} ar^n$ 的级数
等比级数	geometric progression	/dʒiːəˈmetrɪk prəˈɡreʃən/	几何级数的另一种称呼
收敛	convergence	/kənˈvɜːdʒəns/	级数部分和序列有有限极限
无穷级数	infinite series	/ˈɪnfɪnɪt ˈsɪəriːz/	项数无限的级数
幂级数	power series	/ˈpaʊə ˈsɪəriːz/	形如 $\sum_{n=0}^{\infty} a_n x^n$ 的级数
反正切函数	arctangent function	/ɑːkˈtændʒənt ˈfʌŋkʃən/	反正切函数，记作 $\arctan x$

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