Problem 37 - Calculus: Early Transcendentals

Question

The meaning of the decimal representation of a number \( 0.d_1d_2d_3. . . .\) (where the digit \( d_i \) is one of the numbers 0, 1, 2, . . . , 9) is that \( 0.d_1d_2d_3d_4 . . . = \frac {d_1}{10} + \frac {d_2}{10^2} + \frac {d_3}{10^3} + \frac {d_4}{10^4} + . . . \) Show that this series always converges.

Step-by-Step Solution

Answer

The series always converges by the geometric series convergence rule.

Step 1: Understanding the Series

The given decimal number is expressed as an infinite series: \[ 0.d_1d_2d_3 \, \ldots = \frac{d_1}{10} + \frac{d_2}{10^2} + \frac{d_3}{10^3} + \ldots \]where each \(d_i\) is a digit between 0 and 9. We are asked to show that this series always converges.

Step 2: Applying Convergence Criteria for Series

A series \(\sum_{n=1}^{\infty} a_n\) converges if its terms approach zero and the sequence of partial sums \(S_n = \sum_{k=1}^{n} a_k\) has a finite limit. In our series, each term \(\frac{d_i}{10^i}\) clearly approaches zero as \(i\) increases, given that \(d_i\) is a finite digit and the denominator increases exponentially.

Step 3: Observing the Decrease in Subsequent Terms

Each digit \(d_i\) is divided by a power of 10, making subsequent terms in the series smaller by a factor of 1/10. Therefore, \(\frac{d_1}{10} \), \(\frac{d_2}{10^2} \), \(\frac{d_3}{10^3} \), etc., decrease rapidly, ensuring that their sum does not become infinite.

Step 4: Using the Test for Convergence of Geometric Series

Our series is similar in form to a geometric series \(\sum_{n=1}^{fty} ar^{n-1}\) with \(|r| < 1\). Each term \(\frac{d_i}{10^i}\) can be considered as analogous to being multiplied by \(\frac{1}{10}\) at each step. Since \(|r| = \frac{1}{10} < 1\), this series satisfies the convergence criteria for geometric series, thus ensuring convergence.

Step 5: Concluding the Convergence

Since the infinite decimal series \(\frac{d_1}{10} + \frac{d_2}{10^2} + \frac{d_3}{10^3} + \ldots\) satisfies the convergence criteria of a geometric series (due to the decreasing magnitude of terms) and each digit is constrained between 0 and 9, the series always converges to a finite limit.

Key Concepts

geometric series

A geometric series is one where each term is a fixed multiple, or common ratio, of the previous term. It's like a chain reaction where each link is proportionally tied to the last. For instance, in our decimal representation series, each term is divided by 10, making 10 the dividing power. This common ratio \(r\) is less than 1. Think about it:

• The first term is divided by 10.
• The next by 10 squared (100).
• And so on, with each further division increasing the power of 10.

This diminishing factor is what characterizes a geometric series and is pivotal to ensuring its convergence.

infinite series

An infinite series is simply a sequence of numbers that continues endlessly. Imagine writing a list that never stops; this might sound impractical in the real world but is perfectly manageable in mathematics. For any decimal number that doesn't naturally end (like 0.3333...), the sequence of terms representing the number is infinite. Each added term in the series further refines the sum closer to the number represented. \(\sum_{n=1}^{\infty} a_n\) describes our infinite series as these terms progress.

decimal representation

Decimal representation refers to expressing numbers in the base-10 system. It’s how you typically see and understand numbers every day. When dealing with 0.1234, you are looking at a decimal number where each digit after the dot refers to fractions of 10:

• The 1 is \(\frac{1}{10}\).
• The 2 is \(\frac{2}{100}\).
• And it continues...

This fraction format informs the geometric series discussed, where each segment of the fraction decreases exponentially, resulting in a precise value as terms increase. By breaking down a number in this way, a series can easily define, add, and reach the total representation of the decimal.

convergence criteria

To determine if an infinite series converges, we rely on convergence criteria. This is a sort of checklist to see if the sum of terms leads to a fixed and finite value. One major rule is that for any given series \(\sum a_n\):

• The terms must tend towards zero.
• The sequence forming the partial sums must approach a finite limit.

In our decimal series, given that each digit \(d_i\) is restricted to 0-9 and the denominator exponentially grows as 10 to some power, each term becomes smaller and smaller. This shrinking makes each progress minute step in the sum, ensuring that convergence happens against a finite backdrop.