Problem 25 - Calculus: One and Several Variables

Question

Express the decimal fraction \(0 . a_{1} a_{2} \cdots a_{n}\) in sigma notation using powers of \(1 / 10\).

Step-by-Step Solution

Answer

The decimal fraction \(0 . a_{1} a_{2} \cdots a_{n}\) is expressed in sigma notation as \(\Sigma_{i=1}^{n} a_i \times (1 / 10)^i\).

Step 1: Understand the decimal fraction

Look at the decimal fraction \(0 . a_{1} a_{2} \cdots a_{n}\). Each digit \(a_i\) represents a term of the series, with \(a_1\) being the first decimal place, \(a_2\) the second, and so forth up until \(a_n\).

Step 2: Convert to sigma notation

To express this in sigma notation, we consider each digit \(a_i\) as a term of the series. Each term in the series is represented as \(a_i\) times \(1 / 10\) (or 0.1) raised to the power of \(i\). this gives \(a_i \times (1 / 10)^i\).

Step 3: Write the complete series in sigma notation

To denote the complete series, we use the sigma notation, which involves the Greek letter sigma \(\Sigma\). We want to sum all these terms from \(i = 1\) (the first decimal place) to \(n\) (the last decimal place). So, the complete sigma notation becomes \(\Sigma_{i=1}^{n} a_i \times (1 / 10)^i\).

Key Concepts

Decimal Expansions

Decimal expansions are the way we represent numbers using a combination of digits and decimal points.
When you see a number like 0.123, it's a decimal fraction showing parts of a whole number. Each digit in the decimal conveys a different level of precision or place value.
For example:

• The first digit after the decimal point is the tenths place.
• The second is the hundredths place.
• The third is the thousandths place, and so on.

This system is based on powers of 10, where each digit's position determines its value. In the exercise, the decimal fraction is expressed as a series of terms where each digit is multiplied by a power of 1/10.
This is important because converting decimals into a form that's expressed mathematically can make it easier to work with in computations and analyses.

Series Representation

In mathematics, expressing a number as a series lets us see it as a sum of its parts. Especially when dealing with decimals, series representation helps clarify how each individual digit contributes to the whole.
In this context, a series is simply the sum of elements that follow a specific pattern. For the decimal fraction in question, each digit is treated as a unique member of the series.
To represent a decimal number as a series:

• Identify each digit as a separate term.
• Each term is the digit multiplied by its place value.
• Add all these terms to find the series representation.

Through such a series, complex numbers and fractions can be broken down into simpler pieces, making mathematical operations more approachable and clear.

Mathematical Notation

Mathematical notation is a kind of language used to succinctly communicate mathematical ideas. It uses symbols and conventions to express different concepts without writing lengthy sentences.
One of the notations mentioned in the exercise is sigma notation, which uses the Greek letter \( \Sigma \) to indicate summation.
In sigma notation:

• The expression \( \Sigma_{i=1}^{n} \) denotes summation from 1 to \( n \).
• The variable \( i \) represents each term's position in the series, iterating over the specified range.
• Each term follows the format \( a_i \times (1 / 10)^i \), showing how each digit contributes to the total.

This concise way of writing sums allows mathematicians and students to easily manage series of numbers without writing them out in full. By mastering these notations, solving complex mathematical problems becomes much more transparent and less cumbersome.