Problem 56 - A Graphical Approach to College Algebra

Question

It can be shown that $$(1+x)^{n}=1+n x+\frac{n(n-1)}{2 !} x^{2}+\frac{n(n-1)(n-2)}{3 !} x^{3}+\cdots$$ for any real number \(n\) ( not just positive integer values) and any real number \(x\), where \(|x|<1 .\) Use this result to approximate each quantity in Exercises \(53-56\) to the nearest thousandth. $$(1.03)^{0.2}$$

Step-by-Step Solution

Answer

\((1.03)^{0.2} \approx 1.006\).

Step 1: Identify Known Values

We are given the expression \((1+x)^n\). In our problem, we want to approximate \((1.03)^{0.2}\). Here, \(1+x = 1.03\) and \(n = 0.2\). Hence, \(x = 0.03\).

Step 2: Substitute Values into the Binomial Series

Substitute the values \(n = 0.2\) and \(x = 0.03\) into the binomial series expansion: \[(1+x)^{n}=1+nx+\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3+\cdots\] which yields \[(1.03)^{0.2} = 1 + 0.2(0.03) + \frac{0.2(0.2-1)}{2!}(0.03)^2 + \frac{0.2(0.2-1)(0.2-2)}{3!}(0.03)^3 + \cdots\]

Step 3: Calculate Each Term

Calculate the first few terms of the series:- The first term is \(1\).- The second term is \(0.2 \times 0.03 = 0.006\).- The third term is \(\frac{0.2(-0.8)}{2} \times (0.03)^2 = -0.000072\).- The fourth term is \(\frac{0.2(-0.8)(-1.8)}{6} \times (0.03)^3 = 0.000000864\).The higher-order terms become very small and contribute negligibly, so we can stop here.

Step 4: Sum the Series Terms

Add the significant terms from the series expansion:\[1 + 0.006 - 0.000072 + 0.000000864 = 1.005928864\]

Step 5: Round to the Nearest Thousandth

Round the result from Step 4, \(1.005928864\), to the nearest thousandth. The value rounded to three decimal places is \(1.006\).

Key Concepts

Approximation Methods

Approximation methods play a crucial role in mathematics, especially when dealing with complex equations where an exact solution is challenging or impossible to obtain. The binomial series is one of these methods. It helps approximate values by expanding a power expression for real numbers around a small value of the variable.

The idea is to break down the power expression into simpler terms that can be calculated easily. For the binomial series \((1+x)^n\), these terms include linear, quadratic, and higher-degree components. Each term adds more precision to the approximation, but often only a few are needed to get an accurate result

• The first term is always 1, representing the constant part of the binomial expansion.
• As you add more terms, the approximation becomes finer, but after a certain point, contributions from additional terms are negligible.

The balance between computational simplicity and required accuracy is the essence of good approximation methods.

Real Exponents

A real exponent is any number that can take the form of rational numbers, integers, or irrational numbers. Unlike integer exponents, which multiply a base by itself a specific number of times, real exponents involve roots and fractional powers. This requires a different approach for calculation, often involving more advanced calculus or series methods.

The binomial series is particularly useful for approximating expressions involving real exponents, as it allows you to represent powers in terms of a series expansion around a small number. This is especially helpful when the exponent is not an integer. In our example, the exponent 0.2 in \((1.03)^{0.2}\) is crucial:

• It shows how these methods work beyond simple multiplication, as it involves fractional powers.
• Using the binomial series effectively handles computations that would otherwise be complex and time-consuming, especially without a calculator.

Understanding real exponents allows for a deeper appreciation of how and why these approximations work.

Binomial Expansion

The binomial expansion is a method of expressing a binomial raised to any power in terms of the sum of terms. The binomial series is a specific type of binomial expansion that extends its utility to any real exponent, not just positive integers. It is expressed as: \((1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \cdots\)

This formula is significant because it simplifies calculations involving powers of a number that are near unity.

• Each term in the series provides a correction that refines the approximation.
• The factor \(n, n(n-1), n(n-1)(n-2)\) in subsequent terms adjusts the weight each correction contributes, accounting for the influence of higher powers of the exponent.
• The factorial \(!\) in the denominator is crucial for scaling down the higher-power contributions since these terms grow faster.

By leveraging these insights, the binomial expansion transforms complex power expressions into manageable arithmetic, providing a practical approach to approximations.