Problem 65 - Thomas Calculus

Question

Decimal numbers Any real number in the interval \([0,1]\) can be represented by a decimal (not necessarily unique) as $$0 . d_{1} d_{2} d_{3} d_{4} \ldots=\frac{d_{1}}{10}+\frac{d_{2}}{10^{2}}+\frac{d_{3}}{10^{3}}+\frac{d_{4}}{10^{4}}+\cdots$$ where \(d_{i}\) is one of the integers \(0,1,2,3, \ldots, 9 .\) Prove that the series on the right-hand side always converges.

Step-by-Step Solution

Answer

The series is a convergent geometric series, ensuring convergence.

Step 1: Understanding the problem

The task is to prove that the infinite series \( \frac{d_1}{10} + \frac{d_2}{10^2} + \frac{d_3}{10^3} + \ldots \), representing any real number within the interval \([0,1]\) as a decimal, always converges. Here, each \(d_i\) is a digit from 0 to 9.

Step 2: Expressing the series mathematically

The series can be seen as a geometric series: \( S = \sum_{i=1}^{\infty} \frac{d_i}{10^i} \). This is because each term in the series is of the form \( \frac{d_i}{10^i} \) where \(d_i\) is a decimal digit.

Step 3: Using the properties of a geometric series

A geometric series of the form \( \sum_{i=0}^{\infty} ar^i \) converges if the common ratio \(|r| < 1\). In this case, our common ratio is \(\frac{1}{10}\), which satisfies \(|\frac{1}{10}| < 1\). Therefore, the series converges.

Step 4: Bounding the sum of the series

Each digit \(d_i\) is between 0 and 9. Thus each term \( \frac{d_i}{10^i} \) is less than or equal to \(\frac{9}{10^i}\). Since it is a convergent geometric series, the sum of the series is bounded.

Step 5: Confirming convergence

Given the series \(\sum_{i=1}^{\infty} \frac{d_i}{10^i} \), and knowing it can be expressed as a geometric series with a ratio less than 1, we can conclude it converges. Specifically, the convergence is to a real number in \([0,1]\).

Key Concepts

Geometric Series

Geometric series are a fascinating concept in mathematics. They involve sequences where each term is derived by multiplying the previous one by a constant, known as the common ratio. A geometric series is structured in the form \(a, ar, ar^2, ar^3, \dots\). The series keeps adding each term, forever in theory, resulting in an infinite series like \(S = a + ar + ar^2 + ar^3 + \dots\).

Conditions for convergence are crucial. For a geometric series \( \sum_{i=0}^{\infty} ar^i \) to converge, the absolute value of the common ratio \(|r|\) must be less than 1. This ensures the terms grow smaller and smaller, eventually totaling to a finite sum.

For example, consider the series \( 1 + 0.1 + 0.01 + 0.001 + \ldots \). Here, \(a = 1\) and \(r = 0.1\). Since \(r\) is less than 1, this series converges. Understanding this concept reveals why even infinite decimal expansions result in finite real numbers.

Decimal Representation

Decimal representation is a method of expressing numbers in a base-10 numeral system. It's the standard form of writing numbers we use daily. Each position represents a power of 10, moving left or right around the decimal point.

To represent a real number in the interval \([0,1]\) as a decimal, we write it in the form \(0.d_1 d_2 d_3 \ldots \). Each digit \(d_i\) is a number from 0 to 9, indicating the fraction of the respective power of 10. For example, the number \(0.326\) represents \(\frac{3}{10} + \frac{2}{100} + \frac{6}{1000}\).

This approach, known as decimal expansion, provides a neat way to represent real numbers. Yet, it's interesting to note that some numbers can have more than one decimal representation, like \(0.999\ldots = 1\). Such cases arise due to the convergence properties of decimal series.

Real Numbers

Real numbers form a complete and continuous line, containing all rational and irrational numbers. They encompass integers, fractions, and decimals, every value along an infinite, unbroken line that extends from \(-\infty\) to \(\infty\).

Crucially, real numbers can represent both finite decimal numbers and repeating decimals. This means numbers like 0.5 and 1/3 (which is represented by the repeating decimal 0.333...) are real numbers.

The concept of real numbers in the interval \([0,1]\) plays a key role in understanding convergence. Decimal expansions of real numbers within this range can be expressed as infinite series, specifically geometric series as discussed earlier.

• This series always converges to a specific real number.
• The convergence is assured by properties of the geometric series where the common ratio is less than 1.

Understanding real numbers and their representation is fundamental to mastering numerous mathematical concepts and seeing how different areas of math interconnect.