Problem 47 - Calculus

Question

Explain what is wrong with the statement. Since $$\frac{1}{1-x}=1+x+x^{2}+x^{3}+\cdots$$ we conclude that $$\frac{1}{1-2}=1+2+2^{2}+2^{3}+\cdots$$

Step-by-Step Solution

Answer

The series diverges for \( x = 2 \) since \( |x| > 1 \).

Step 1: Recognize the Series

The statement given uses a mathematical series that sums to the function \( \frac{1}{1-x} \). This series is known as the geometric series: \( 1 + x + x^2 + x^3 + \cdots \).

Step 2: Identify the Validity Range

The geometric series \( 1 + x + x^2 + x^3 + \cdots \) converges to \( \frac{1}{1-x} \) only when the absolute value of \( x \) is less than 1, i.e., \( |x| < 1 \).

Step 3: Identify the Error

The error in the statement is the calculation \( \frac{1}{1-2}=1+2+2^{2}+2^{3} + \cdots \), which uses \( x = 2 \). Since \( |2| > 1 \), the series does not converge and the geometric series formula cannot be used.

Step 4: Conclusion

The convergence condition \( |x| < 1 \) is violated, making the series diverge. Hence, \( \frac{1}{1-2} \) cannot be represented by this series.

Key Concepts

Convergence: Understanding Convergence of a Geometric Series

Geometric series play a significant role in mathematics. They are infinite series of the form \( 1 + x + x^2 + x^3 + \cdots \) and can be summed up to the expression \( \frac{1}{1-x} \) under certain conditions. For convergence to occur, the series must approach a finite value as more terms are added. This is a key condition for the geometric series.The most important requirement for the geometric series \( 1 + x + x^2 + x^3 + \cdots \) to sum up to \( \frac{1}{1-x} \) is that the absolute value of \( x \) is less than 1. This means:

• The variable \( x \) should lie between \(-1\) and \(1\).
• If \( x \) meets this criterion, you can confidently say the series converges to \( \frac{1}{1-x} \).

If \( |x| < 1 \), adding more terms of the series will yield a sum that gets closer and closer to a fixed number. This represents the idea of convergence telling you that under this condition, the series provides a valid and meaningful arithmetic result.

Divergence: Understanding Divergence in a Geometric Series

When a geometric series does not converge, it diverges. Divergence implies that as terms are added, the series does not approach a finite sum. Instead, the sum either increases indefinitely or oscillates without settling around a particular value. Consider the geometric series \( 1 + 2 + 2^2 + 2^3 + \cdots \). Here, \( x = 2 \), and since \( |x| > 1 \), this series diverges.

• If \(|x| \geq 1\), the terms of the series do not decrease in magnitude, leading to divergence.
• This lack of convergence means you cannot represent the series with a simple formula like \( \frac{1}{1-x} \).

Divergence often acts as a warning that a series, like our example with \( x = 2 \), goes beyond the limits suitable for simple summation formulas, as the terms become overwhelmingly large.

Series Validity Range: Determining the Range for Valid Series Representation

In understanding both convergence and divergence, the series validity range is crucial. For the geometric series \( 1 + x + x^2 + x^3 + \cdots \), the validity range is directly tied to the convergence criterion of \( |x| < 1 \).This range involves:

• Checking \( |x| < 1 \): This ensures the series will converge, making \( \frac{1}{1-x} \) a valid expression for the sum.
• If \( x \) falls outside the range \(|x| < 1\), the series does not offer a valid sum.

The summary of validity is simple: when \(|x| \geq 1\), the series goes beyond this designated validity range, leading to results that don't make mathematical sense. Therefore, understanding this range helps you apply geometric series accurately and avoid errors in calculations, like attempting to calculate \( \frac{1}{1-2} \) using an incorrect series framework.