Question

Robert wants to know his body mass to arbitrary precision. He has a big balancing scale that works perfectly, an unlimited collection of \(1-\mathrm{kg}\) weights, and nine each of \(0.1-\mathrm{kg}, 0.01-\mathrm{kg}, 0.001-\mathrm{kg}\), and so on weights. Assuming the scale is big enough, can he do this? What does this have to do with infinite series?

Short answer

Yes, Robert can achieve this using the concept of a geometric series.

Step-by-step solution

Understanding the Problem

Robert wants to measure his body mass using a balance scale and a collection of weights with progressively smaller masses: one quantity of 1-kg weights, and nine of each smaller unit down to any precision. This means he aims to use the scale to reach an accurate measurement close to his true mass to any desired precision.

Setting Up Weights as a Series

The weights Robert can use form a sequence where each subsequent weight is a fraction (1/10th) of the previous smaller weight: 1 kg, 0.1 kg, 0.01 kg, 0.001 kg, etc. This mimics the structure of a series where each term decreases exponentially by a factor of 1/10.

Recognizing a Geometric Series

The weights can be represented as a geometric series with initial term 1 and common ratio 1/10, expressed as: \[ 1 + \frac{1}{10} + \frac{1}{100} + \frac{1}{1000} + \ldots \] Each term is measured precisely by a smaller unit of mass.

Calculating the Sum of the Series

The sum of an infinite geometric series can be calculated using the formula: \[ S = \frac{a}{1 - r} \] where \( a \) is the first term of the series (1 kg), and \( r \) is the common ratio (1/10). The series converges to \( \frac{1}{0.9} = \frac{10}{9} \), but practically each finite approximation uses smaller units to get arbitrarily close to Robert's precise weight.

Application of Series to Weighing

By using progressively smaller and smaller weights that can be added incrementally, Robert can adjust the total mass on the scale. Since he has enough smaller weights for each term of a geometric series, he can sum the series to get increasingly accurate measurements approaching his true mass.

Conclusion and Connection to Infinite Series

Robert can use the weights to measure his body mass with any required precision. This approach works because he can utilize each term in the geometric series to incrementally approximate his mass, effectively 'summing' the series through physical weights. Infinite series concepts show how approximations can get arbitrarily close to a total mass, aligning with his goal.

Key concepts

Understanding Geometric Series

Geometric series are sequences of numbers where each term is a fixed multiple of the previous one. In Robert's exercise, he utilized weights that decrease exponentially. He starts with a 1 kg weight and measures using weights that are ten times smaller each time: 0.1 kg, 0.01 kg, 0.001 kg, etc.
This structure forms a geometric series. The first term in Robert's weight series is 1, and the common ratio is 1/10. The series is written as:
\[ 1, \frac{1}{10}, \frac{1}{100}, \frac{1}{1000}, \ldots \]
Such a series is useful because it can theoretically add up to an infinite horizon while adding smaller and smaller amounts, thus closely approximating any value thoroughly. Understanding this series helps visualize how Robert can use weights to precisely match his body mass.

Precision in Measurements

Measurement precision is about achieving the closest possible result to the exact value. Robert desired a weight measurement that could be refined indefinitely, i.e., he wanted more digits of precision.
The progression of smaller weighted units allows him to achieve this. For every smaller weight added, the scale adjusts slightly closer to the target mass. In practical terms, this means Robert can improve the accuracy of his measurement without limit, provided he has sufficient smaller weights and time to adjust them correctly. This concept of precision with bounds highlights mathematics' power in tangible real-world scenarios.

Convergence of Series

Convergence describes how a series approaches a stable limit. When dealing with an infinite geometric series, we use the formula for sum convergence given by:

• \( S = \frac{a}{1 - r} \)
• Here, \( a \) is the first term, and \( r \) is the common ratio.

For Robert's problem, this equates to \[ \frac{1}{1 - \frac{1}{10}} = \frac{10}{9} \]
This means the sum of the series converges towards \( \frac{10}{9} \). However, in practice, each additional weight measurement gets his total measurement closer to his actual body mass. The concept of convergence is essential as it defines how accurately and closely we can approach an exact value using mathematics, proving the theoretical underpinnings of Robert's precise mass measurement.

Mathematics in Physical Applications

The application of mathematics in the physical world, like in Robert's situation, demonstrates the practicality and necessity of abstract mathematical concepts. Robert used geometric series to solve a real-life problem—measuring his body mass accurately.
By understanding and applying the convergence of an infinite series, he could make precise measurements reflecting his true body mass, showcasing the typical use of theoretical mathematics in everyday applications.
Such mathematical methods are not limited to weight measurements; they extend to various fields such as engineering and physics, where precise measurements and approximations are crucial.

• Allows for complex calculations.
• Ensures high precision in scientific inquiries.
• Enables accurate modeling and predictions.

This example reveals how mathematical theory translates into tangible, useful solutions, pushing boundaries beyond theoretical exercises into substantial, real-world benefits.