write out the first 5 terms of the Binomial series with the given k-value.
k = \(\frac{1}{2}\)
We can define the binomial series formula as \(\sum_{n = 0}^{inf} \binom{k}{n} x^n\)
or
\(\sum_{n = 0}^{inf} \binom{(1/2)}{n} x^n\)
And of course, to solve for \(\binom{k}{n}\) we can use the formula:
\(\frac{k!}{n!(k - n)!}\)
And from there, we can just substitute n with each iteration in the series:
#Term 1
print(factorial((1/2)) / (factorial((1/2)- 0))) ## [1] 1#Term 2
print(factorial((1/2)) / (factorial(1) * factorial((1/2)- 1))) ## [1] 0.5#Term 3
print(factorial((1/2)) / (factorial(2) * factorial((1/2)- 2)))## [1] -0.125#Term 4
print(factorial((1/2)) / (factorial(3) * factorial((1/2)- 3))) ## [1] 0.0625#Term 5
print(factorial((1/2)) / (factorial(4) * factorial((1/2)- 4))) ## [1] -0.0390625