{Use of Tech} Binomial series


b. Use the first four terms of the series to approximate the given quantity.


f(x) = (1+x)⁻²/³; approximate 1.18⁻²/³.

Verified step by step guidance

1. Identify the function and the point of expansion: The function is given as \(f(x) = (1 + x)^{-\frac{2}{3}}\). We want to approximate \(1.18^{-\frac{2}{3}}\). Notice that \(1.18\) can be written as \(1 + 0.18\), so we will use \(x = 0.18\) for the binomial series expansion around \(x=0\).
2. Recall the binomial series expansion formula for \((1 + x)^n\) where \(n\) is any real number: \[ (1 + x)^n = 1 + n x + \frac{n (n - 1)}{2!} x^2 + \frac{n (n - 1) (n - 2)}{3!} x^3 + \cdots \] Here, \(n = -\frac{2}{3}\).
3. Write out the first four terms of the binomial series for \(f(x) = (1 + x)^{-\frac{2}{3}}\) explicitly: \[ f(x) \approx 1 + n x + \frac{n (n - 1)}{2} x^2 + \frac{n (n - 1) (n - 2)}{6} x^3 \] Substitute \(n = -\frac{2}{3}\) into each coefficient.
4. Calculate each coefficient step-by-step: - First term is always 1. - Second term coefficient is \(n = -\frac{2}{3}\). - Third term coefficient is \(\frac{n (n - 1)}{2}\). - Fourth term coefficient is \(\frac{n (n - 1) (n - 2)}{6}\). Evaluate these expressions symbolically without plugging in the decimal value yet.
5. Substitute \(x = 0.18\) into the four-term approximation and write the expression for the approximate value of \(1.18^{-\frac{2}{3}}\) as: \[ f(0.18) \approx 1 + n (0.18) + \frac{n (n - 1)}{2} (0.18)^2 + \frac{n (n - 1) (n - 2)}{6} (0.18)^3 \] This expression can then be evaluated to approximate the value.

Key Concepts

Binomial Series Expansion

The binomial series generalizes the binomial theorem to any real exponent, allowing the expansion of expressions like (1 + x)^n into an infinite series. For |x| < 1, it is expressed as 1 + n x + n(n-1)/2! x^2 + ..., which helps approximate functions that are difficult to compute directly.

Using the First Few Terms for Approximation

Approximating a function using the first few terms of its series expansion provides a practical way to estimate values without calculating the entire infinite series. The accuracy depends on the number of terms used and the size of x; typically, the first four terms give a reasonable approximation for small x.

Substitution and Simplification in Series

To approximate a specific value like 1.18^(-2/3), rewrite it in the form (1 + x)^n with x = 0.18 and n = -2/3. Then substitute x and n into the binomial series terms, calculate each term, and sum them to find the approximate value, simplifying the process of evaluating complex powers.