Problem 39 - Essential Calculus Early Transcendentals

Question

The meaning of the decimal representation of a number $$0 . d_{1} d_{2} d_{3} \ldots\( (where the digit \)d_{i}\( is one of the numbers \)0,1\( \)2, \ldots, 9 )\( is that \)\quad 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$$ Show that this series always converges.

Step-by-Step Solution

Answer

The series converges because it is bounded by a convergent geometric series.

Step 1: Understanding Convergence

To determine if the series converges, we need to check if the series has a finite limit as the number of terms increases to infinity. A series of real numbers \(a_n\) converges if the sequence of partial sums \(S_n = a_1 + a_2 + \ldots + a_n\) tends to a finite limit.

Step 2: Expressing the Decimal Representation

The given decimal number is expressed as a series: \(0.d_1d_2d_3\ldots = \frac{d_1}{10} + \frac{d_2}{10^2} + \frac{d_3}{10^3} + \ldots\), where each digit \(d_i\) is between 0 and 9.

Step 3: Consider the General Term

Each term of the series can be written as \(a_i = \frac{d_i}{10^i}\). This means the general term of the series is \(\frac{d_i}{10^i}\).

Step 4: Boundedness of the Terms

Notice that for each digit \(d_i\), \(0 \leq d_i \leq 9\). Therefore, \(\frac{d_i}{10^i} \leq \frac{9}{10^i}\). Each term is bounded above by \(\frac{9}{10^i}\).

Step 5: Applying the Comparison Test

To show convergence, use the comparison test. The series \(\sum_{i=1}^{\infty} \frac{9}{10^i}\) is a geometric series with common ratio \(r = \frac{1}{10}\), which is less than 1.

Step 6: Evaluating the Bounding Geometric Series

The series \(\sum_{i=1}^{\infty} \frac{9}{10^i}\) is a geometric series where each term is \(\frac{9}{10^i}\). The first term (\(a\)) is \(\frac{9}{10}\) and the common ratio (\(r\)) is \(\frac{1}{10}\). The sum of an infinite geometric series \(\frac{a}{1-r}\) converges to \(\frac{\frac{9}{10}}{1-\frac{1}{10}} = \frac{9}{10} \times \frac{10}{9} = 1\).

Step 7: Conclusion on Convergence

Since the original series is bounded above by a convergent series, it must also converge. Therefore, the series \(\sum_{i=1}^{\infty} \frac{d_i}{10^i}\) converges for any choice of \(d_i\).

Key Concepts

Decimal Representation

The concept of decimal representation is essential for understanding how numbers are expressed using decimals. A decimal number such as 0.456 is written in a way where each digit has a specific placement value, contributing to the whole number in different magnitude levels:

• The first digit to the right of the decimal point represents the tenths place.
• The second digit represents the hundredths place, and so on.

Each digit after the decimal can be considered as a fraction with a denominator that's a power of 10. For instance, for the number 0.456, it can be broken down as follows:

• 0.4, which is \( \frac{4}{10} \)
• 0.05, which is \( \frac{5}{100} \)
• 0.006, which is \( \frac{6}{1000} \)

When combined, these fractions total to the original decimal number, 0.456. Understanding this breakdown is vital since it allows decimal numbers to be represented as an infinite series where each term diminishes in impact as you move further from the decimal point.

Geometric Series

A geometric series is one where each term is based on multiplying the preceding term by a fixed, non-zero number known as the common ratio. For example, in the series \(1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots\), the common ratio is \(\frac{1}{2}\). This type of series can be finite or infinite:

• In a finite geometric series, the number of terms is limited.
• In an infinite geometric series, it contains endless terms.

The sum of an infinite geometric series is calculated by the formula \( \frac{a}{1-r} \), where \( a \) is the first term and \( r \) is the common ratio. The series converges only when the absolute value of the common ratio \( \lvert r \rvert < 1 \).
In practical applications, this type of series helps determine whether certain decimal representations, like those seen in the original problem's series \( \sum_{i=1}^{\infty} \frac{9}{10^i} \), will have a finite sum. Since the common ratio is \(\frac{1}{10}\), which is less than 1, this series converges, having a predictable finite sum.

Comparison Test

The comparison test is a powerful tool in mathematics to determine the convergence or divergence of an infinite series. It involves comparing the series of interest with another known series:

• If the series is smaller term-by-term compared to a known convergent series, then it also converges.
• If the series is larger term-by-term compared to a known divergent series, then it diverges.

This method simplifies the complex task of determining convergence by comparing to simple series with known behavior. When applied to series of decimal representations, such as in the given problem \( \sum_{i=1}^{\infty} \frac{d_i}{10^i} \), it helps show convergence since each term \( \frac{d_i}{10^i} \) is smaller than \( \frac{9}{10^i} \).
The comparison with the geometric series \( \sum_{i=1}^{\infty} \frac{9}{10^i} \), which converges due to the common ratio being \(\frac{1}{10}\), implies our original series is also convergent. This confirms that any potential decimal constructing this series will represent a real, finite number.