In Exercises 21-24, write out the first 5 terms of the Binomial series with the given k-value.
k = 1/2
\(f(x) = (1+x)^{k}\)
\(k^{0}x^{0} + k^{1}x^{1} + \frac{k(k-1)}{2!}x^2 + \frac{k(k-1)(k-2)}{3!}x^3 + \frac{k(k-1)(k-2)(k-3)}{4!}x^4\)
\((1\times1) + (\frac{1}{2}x) + (\frac{\frac{1}{2}(\frac{1}{2}-1)}{2!}x^2) + (\frac{\frac{1}{2}(\frac{1}{2}-1)(\frac{1}{2}-2)}{3!}x^3) + (\frac{\frac{1}{2}(\frac{1}{2}-1)(\frac{1}{2}-2)(\frac{1}{2}-3)}{4!}x^4)\)
\(1 + (\frac{1}{2}x) + (\frac{\frac{1}{2}(-\frac{1}{2})}{2!}x^2) + (\frac{\frac{1}{2}(-\frac{1}{2})(-1\frac{1}{2})}{3!}x^3) + (\frac{\frac{1}{2}(-\frac{1}{2})(-1\frac{1}{2})(-2\frac{1}{2})}{4!}x^4)\)
\(1 + (\frac{1}{2}x) + (\frac{\frac{1}{2}(-\frac{1}{2})}{2\times1}x^2) + (\frac{\frac{1}{2}(-\frac{1}{2})(-1\frac{1}{2})}{3\times2\times1}x^3) + (\frac{\frac{1}{2}(-\frac{1}{2})(-1\frac{1}{2})(-2\frac{1}{2})}{4\times3\times2\times1}x^4)\)
\(1 + (\frac{1}{2}x) + (\frac{-\frac{1}{4}}{2}x^2) + (\frac{\frac{3}{8}}{6}x^3) + (\frac{-\frac{15}{16}}{24}x^4)\)
\(1 + (\frac{1}{2}x) + (-\frac{1\times2}{4}x^2) + (\frac{3\times6}{8}x^3) + (-\frac{15\times24}{16}x^4)\)
\(1 + (\frac{1}{2}x) + (-\frac{2}{4}x^2) + (\frac{18}{8}x^3) + (-\frac{360}{16}x^4)\)
\(1 + (\frac{1}{2}x) + (-\frac{1}{2}x^2) + (2\frac{2}{8}x^3) + (-22\frac{8}{16}x^4)\)
\(1 + (\frac{1}{2}x) + (-\frac{1}{2}x^2) + (2\frac{1}{4}x^3) + (-22\frac{1}{2}x^4)\)
\(1 + \frac{x}{2} - \frac{x^2}{2} + 2\frac{x^3}{4} - 22\frac{x^4}{2}\)