Problem 53 - 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. $$(1.02)^{-3}$$

Step-by-Step Solution

Answer

Approximately 0.942.

Step 1: Identify the Values

We have the expression \((1+x)^{n}\) where \(x = 0.02\) and \(n = -3\). Use these in the binomial series expansion to approximate \((1.02)^{-3}\).

Step 2: Apply Binomial Expansion Formula

Using the binomial series expansion, substitute \(n = -3\) and \(x = 0.02\) into the expression:\[(1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \cdots\]

Step 3: Calculate the First Term

The first term is 1, since it is \[1\]

Step 4: Calculate the Second Term

The second term is obtained by multiplying \(n = -3\) and \(x = 0.02\):\[-3 \times 0.02 = -0.06\]

Step 5: Calculate the Third Term

The third term is given by \[ \frac{-3(-3-1)}{2}(0.02)^2 = \frac{-3(-4)}{2}(0.02)^2 = \frac{12}{2}(0.0004) = 6 \times 0.0004 = 0.0024\]

Step 6: Calculate the Fourth Term

The fourth term is \[ \frac{-3(-4)(-5)}{6}(0.02)^3 = \frac{60}{6}(0.000008) = 10 \times 0.000008 = 0.00008\]

Step 7: Sum the Terms

Add all calculated terms together to approximate the expression:\[1 - 0.06 + 0.0024 - 0.00008\]

Step 8: Final Calculation

Perform the arithmetic:\[1 - 0.06 = 0.94\]\[0.94 + 0.0024 = 0.9424\]\[0.9424 - 0.00008 = 0.94232\]

Step 9: Round to the Nearest Thousandth

Round the final approximation to the nearest thousandth:\[0.94232 \approx 0.942\]

Key Concepts

Real Numbers

Real numbers include all the numbers we usually think of from our everyday lives, such as the whole numbers, fractions, and decimals. They can be positive, negative, or zero. Real numbers are important here because they allow us to handle a broader set of values in the binomial series expansion.
Unlike integers, which are whole numbers, real numbers fill in all the spaces between integers, providing a continuous sequence.

• Whole numbers like 3, 0, and -1.
• Decimals like 4.56 and -0.73.
• Fractions such as \(\frac{1}{2}\) and -\(\frac{5}{3}\).

In this exercise, both the exponent \(n\) and the variable \(x\) are real numbers. Specifically, the value of \(n = -3\) attributes to a negative real number. This highlights the flexibility of using real numbers, which can conveniently represent scenarios not possible with other number sets, like integers.

Series Expansion

Series expansion is the technique of expressing a function as an infinite sum of terms. In mathematics, series are a concise format to approximate complex functions using simpler polynomials or recognizable terms for easier computation.
The binomial series provides a powerful tool for expanding the expression \((1 + x)^n\) into a more digestible form.

• The expansion starts with 1 and subsequent terms involve powers of \(x\).
• Each term in the series provides a more accurate approximation by capturing the contribution of higher powers of \(x\).
• The coefficients of these terms are derived from the binomial coefficients found in Pascal's triangle.

The formulation used here applies these principles: \[(1+x)^{n}=1+n x+\frac{n(n-1)}{2 !} x^{2}+\frac{n(n-1)(n-2)}{3 !} x^{3}+ \cdots\]This lets us express expressions like \((1.02)^{-3}\) in a straightforward, computable format that leverages the real power of series expansion in handling real numbers.

Approximation Techniques

Approximation techniques involve using mathematical tools and methods to find an estimated value that is close enough to the exact value, usually when direct calculation is too complex. In this exercise, the binomial series is used as an approximation technique, facilitating a simpler understanding of complex expressions.
The series expansion can be truncated to include only a few terms, providing a good approximation.

• By calculating just the first few terms, \(1, -0.06, 0.0024, -0.00008\), we get a very close approximation of \((1.02)^{-3}\).
• The power of approximation is that it offers accuracy with a minimal number of calculations, indispensable for real-world computations.
• In practice, limiting the number of terms uses helps manage complexity, and much of mathematics involves striking this balance for practical applications.

By summing these calculated terms and rounding to the nearest thousandth, we derive a simplified form that reflects the essence of approximation: finding a balance between simplicity and precision. This approach makes mathematics incredibly useful for approximating real-world phenomena accurately and efficiently.