Question

{Use of Tech} Binomial seriesb. Use the first four terms of the series to approximate the given quantity.f(x) = (1+x)⁻²/³; approximate 1.18⁻²/³.

Accepted Answer

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$. Recall the binomial series expansion formula for (1 + x)^n$ 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$$ = -\frac{2}{3}$$. 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. 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. 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.

{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.

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Key Concepts

Binomial Series Expansion

Using the First Few Terms for Approximation

Substitution and Simplification in Series