Problem 84 - Essential Calculus

Question

Consider the formula \(\frac{1}{x-1}=1+x+x^{2}+x^{3}+\cdots\) Given \(x=-1\) and \(x=2\), can you conclude that either of the following statements is true? Explain your reasoning. (a) \(\frac{1}{2}=1-1+1-1+\cdots\) (b) \(-1=1+2+4+8+\cdots\)

Step-by-Step Solution

Answer

No, neither statement (a) nor statement (b) is true. When you plug \(x=-1\) into the equation, you get \(-\frac{1}{2}\), not \(\frac{1}{2}\), refuting statement (a). In the case of \(x=2\), you get \(1\), not \(-1\), thereby refuting statement (b).

Step 1: Examination of statement (a)

Plugging \(x=-1\) into the equation \(\frac{1}{x-1}\) results in \(\frac{1}{-1-1}\), which simplifies to \(-\frac{1}{2}\). Notice, however, that the sum given in statement (a) is \(\frac{1}{2}\), not \(-\frac{1}{2}\). Thus, statement (a) is not correct.

Step 2: Examination of statement (b)

Now consider \(x=2\). Plugging this into the equation \(\frac{1}{x-1}\) gives \(\frac{1}{2-1}\), which simplifies to \(1\). Nonetheless, the sum given in statement (b) is \(-1\), not \(1\). Consequently, statement (b) is not correct either.

Key Concepts

Mathematical Series

In mathematics, a series is the sum of the terms in a sequence. Think of it like adding up a list of numbers, but sometimes this list can be infinitely long. A good example is the series given in the exercise, where numbers are formed by repeating a certain operation (like multiplying by the same number) endlessly. The simplicity of a mathematical series often disguises its complexity, especially when we're dealing with infinite sequences.

It's important to remember that not all series neatly sum up to a single number. In some cases, like with the series presented for the values of x in the exercise, determining the sum can get quite tricky. That's why understanding the convergence or divergence of series is an essential skill in math, especially in calculus and higher-level math courses. By breaking down series into more understandable parts and using algebraic manipulation, we can unravel the underlying patterns and attempt to discover their sum.

Divergent Series

When talking about infinite series, we often want to know if the series converges or diverges. A divergent series is essentially a series that doesn't have a finite sum. That means no matter how many terms you add up, the total will keep growing indefinitely, never settling at a specific number. The series given in statement (b) of the exercise, 1 + 2 + 4 + 8 + ..., when x=2, is an example of a divergent series.

Despite its inability to sum to a finite number, understanding divergent series helps us in various mathematical analyses, such as in the study of Fourier series or in understanding the behavior of functions. So, even though they may seem like mathematical oddities, divergent series have their place in the grand scheme of math.

Algebraic Manipulation

At its core, algebraic manipulation is the process by which we transform equations and expressions into different, often simpler, forms. It involves the use of algebraic properties, like the distributive, associative, and commutative rules, to rearrange and combine like terms. The exercise demonstrates this in how we substitute x with specific values to transform the general form into a particular series.

Correct manipulation is the backbone of algebra and is crucial for solving equations, simplifying expressions, and understanding series convergence. Though it may start simple, like combining like terms or factoring polynomials, it can get more sophisticated with series and higher math. By becoming adept with these algebraic tools, students can solve not only textbook exercises but also real-world problems that involve complex mathematical modeling.