Question

Explain why, if \(x>1 / 2\), then \(x\) cannot be written \(x=\sum_{n=2}^{\infty} \frac{b_{n}}{2^{n}}\left(b_{n}=0\right.\) or \(\left.1, b_{1}=0\right)\).

Short answer

If \( x > 1/2 \), the series \( x = \sum_{n=2}^{\infty} \frac{b_{n}}{2^{n}} \) cannot exceed 1, hence \( x \) cannot be represented as lower than or equal to \( 1/2 \) with \( b_1 = 0 \).

Step-by-step solution

Analyze the Given Series

The problem states that if \(x = \sum_{n=2}^{\infty} \frac{b_{n}}{2^{n}}\), where each \(b_n\) is either 0 or 1, and \(b_1 = 0\), then this series represents a binary fraction starting with the term for \(n=2\). Here, we need to understand that \(x > \frac{1}{2}\).

Understand the Series Representation

In a binary fraction representation, each coefficient \(b_n\) represents a power of \(2\), with subsequent powers becoming smaller as \(n\) increases. The maximum value the series can reach would be if all coefficients except \(b_1\) are 1, yielding \(x = \sum_{n=2}^{\infty} \frac{1}{2^n} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots\).

Calculate the Limit of the Series

Compute the sum of the series for \(x\): \[ x = \sum_{n=2}^{\infty} \frac{1}{2^n} = \frac{1}{2} \left(\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots\right) \]This is an infinite geometric series with the first term \(a=\frac{1}{2}\) and a common ratio \(r=\frac{1}{2}\).

Use the Formula for Infinite Geometric Series

Recall the formula for the sum of an infinite geometric series: \[ \text{Sum} = \frac{a}{1 - r}\]Substitute \(a = \frac{1}{2}\) and \(r = \frac{1}{2}\):\[ \text{Sum} = \frac{1/2}{1 - 1/2} = 1\]Thus, the sum of this binary representation adds up to 1.

Compare the Series Sum with the Given Condition

Since the sum of the series equals 1, \(x\) cannot exceed 1. This implies that if any partial sum of the binary series starts exceeding \(\frac{1}{2}\), the series with all \(b_n\) being either 0 or 1 cannot satisfy a condition where \(x > \frac{1}{2}\) with \(b_1 = 0\). The possible range for \(x\) given this series is up to, but not beyond, 1.

Key concepts

Binary Fraction

Binary fractions are a way of representing numbers in base-2 format. Just like decimal fractions use powers of ten, binary fractions use powers of two to express values.
Each digit in a binary fraction (0 or 1) represents a power of two, with larger exponents corresponding to digits further to the left.

• The first position to the right of the binary point represents \( \frac{1}{2} \).
• The second position represents \( \frac{1}{4} \).
• As we move further right, each subsequent position represents half the value of the previous one. For example, \( \frac{1}{8} \), \( \frac{1}{16} \), and so on.

This setup allows us to convert any decimal fraction into a binary fraction by determining which powers of two sum up to give the fractional part of the number. In the context of the series given, the binary fraction begins at the second term due to the condition \( b_1 = 0 \), affecting the possible values for the sum of the series.

Series Representation

In mathematics, a series is a way of expressing a sum of terms from a sequence. Representing a number as a series of terms allows us to understand how different units contribute to the whole.
The exercise we're examining considers a specific series where each term is a binary fraction, represented by powers of \(2\).

• The notation \(x = \sum_{n=2}^{\infty} \frac{b_{n}}{2^{n}}\) means we are summing an infinite series of terms, each divided by an increasing power of two.
• The index "n" begins at 2 in this scenario because \(b_1\) is constrained to be 0.

The integer values of \(b_n\) (either 0 or 1) determine which powers of two are present in the sum. As each term is added to the sequence, it incrementally contributes to building the representation of the number \(x\).

Geometric Series Formula

A geometric series is one where each term is a fixed multiple of the previous term. When the series is infinite and the common ratio is less than one in absolute value, it converges to a specific number. This convergence is simplified by using the formula for the sum of an infinite geometric series.
The general formula for the sum \( S \) of an infinite geometric series is:
\[ S = \frac{a}{1 - r} \]
where \(a\) represents the first term and \(r\) is the common ratio of the series.

• For our specific series, \(a = \frac{1}{2}\) because the series starts at \( \frac{1}{2} \).
• The common ratio \(r\) in this case is also \( \frac{1}{2} \), since every subsequent term is halved.

Using the formula, we find that the series converges to 1. This result elucidates why numbers represented by this type of series never exceed 1, even if they can get very close. The series cannot reach a sum greater than 1 because adding powers of 2 greater than zero (like \(1/4, 1/8, etc.\)) results in progressively smaller additions, keeping the sum bounded.