#In Exercises 21 – 24, write out the first 5 terms of the Binomial series with the given k-value.
21. \(k = \frac{1}{2}\):
The Binomial series with \(k = \frac{1}{2}\) expands as:
\[ (1 + x)^{\frac{1}{2}} = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 - \frac{5}{128}x^4 + \cdots \]
So, the first five terms are:
\[ 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 - \frac{5}{128}x^4 \]
22. \(k = -\frac{1}{2}\):
The Binomial series for \((1 + x)^{-\frac{1}{2}}\) expands as:
\[ (1 + x)^{-\frac{1}{2}} = 1 - \frac{1}{2}x - \frac{1}{8}x^2 - \frac{1}{16}x^3 - \frac{5}{128}x^4 - \cdots \]
So, the first five terms are:
\[ 1 - \frac{1}{2}x - \frac{1}{8}x^2 - \frac{1}{16}x^3 - \frac{5}{128}x^4 \]
23. \(k = \frac{1}{3}\):
The Binomial series for \((1 + x)^{\frac{1}{3}}\) expands as:
\[ (1 + x)^{\frac{1}{3}} = 1 + \frac{1}{3}x - \frac{1}{9}x^2 + \frac{5}{81}x^3 - \frac{5}{243}x^4 + \cdots \]
So, the first five terms are:
\[ 1 + \frac{1}{3}x - \frac{1}{9}x^2 + \frac{5}{81}x^3 - \frac{5}{243}x^4 \]
24. \(k = 4\):
The Binomial series for \((1 + x)^{4}\) expands as:
\[ (1 + x)^{4} = 1 + 4x + 6x^2 + 4x^3 + x^4 + \cdots \]
So, the first five terms are:
\[ 1 + 4x + 6x^2 + 4x^3 + x^4 \]