Write out the first five terms of the Binomial series with the given k-values
The binomial series expansion for the first five terms for any value of \(k \neq 0\) is as follows:
\[ (1+x)^k = 1 + kx + \frac{k(k-1)}{2!}x^2 + \frac{k(k-1)(k-2)}{3!}x^3 + \frac{k(k-1)(k-2)(k-3)}{4!}x^4\]
\[ (1+x)^k = 1 + \frac{1}{2}x + \frac{\frac{1}{2}(\frac{1}{2}-1)}{2!}x^2 + \frac{\frac{1}{2}(\frac{1}{2}-1)(\frac{1}{2}-2)}{3!}x^3 + \frac{\frac{1}{2}(\frac{1}{2}-1)(\frac{1}{2}-2)(\frac{1}{2}-3)}{4!}x^4\]
The first five terms when k = \(\frac{1}{2} =\)
\[ (1+x)^k = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 - \frac{5}{128}x^4\]
\[ (1+x)^k = 1 - \frac{1}{2}x + \frac{-\frac{1}{2}(-\frac{1}{2}-1)}{2!}x^2 + \frac{-\frac{1}{2}(-\frac{1}{2}-1)(-\frac{1}{2}-2)}{3!}x^3 + \frac{-\frac{1}{2}(-\frac{1}{2}-1)(-\frac{1}{2}-2)(-\frac{1}{2}-3)}{4!}x^4\]
The first five terms when k = \(-\frac{1}{2}=\)
\[ (1+x)^k = 1 - \frac{1}{2}x + \frac{3}{8}x^2 - \frac{5}{16}x^3 + \frac{35}{128}x^4 \]