Write out the first 5 terms of the Binomial series with the given k-value.
k = 1/2
Christian’s Response:
The Binomial Series formula is as follows
\[ (1 + x)^k = 1 + kx + (k * (k-1) / 2!) x^2 + (k * (k-1) * (k-2) / 3!) x^3 + ... \]
k is our constant and x is a variable. To solve, we substitute k with 1/2
First Term: Our first term will always be 1
Second Term: The second term kx will become (1/2)x
Third Term: The third term (k*(k-1)/2!)x^2 can be broken down to (1/8)x^2
Fourth Term: The fourth term (k(k-1)(k-2)/3!)x^3 can be broken down to (1/16)x^3
Fifth Term: The fifth term (k(k-1)(k-2)*(k-3)/4!)x^4 can be broken down to -(5/128)x^4
The first five terms in the Binomial Series is
\[ 1 + (1/2)x - (1/8)x^2 + (1/16)x^3 - (5/128)x^4 \]