DATA 605 Week 14 Discussion

INSTRUCTIONS

In Exercises 21 – 24 of APEX Calculus V 4.0, write out the first 5 terms of the Binomial series with the given k-value.

\(k = 4\)

SOLUTION

The Binomial series is given by:

\((1 + x)^{k} = \sum_{n=0}^{\infty} {k \choose n} (x)^{n}\)

where \({k \choose n}\) is the binomial coefficient, defined as \({k \choose n} = \frac{k!}{n!(k-n)!}\).

For \(k=4\), the first 5 terms of the Binomial series are as follows:

\({4 \choose 0} (x)^{0} + {4 \choose 1} (x)^{1} + {4 \choose 2} (x)^{2} + {4 \choose 3} (x)^{3} + {4 \choose 4} (x)^{4}\)

Simplifying the binomial coefficients, we get the following:

\(1 + 4x + 6(x)^{2} + 4x^{3} + x^{4}\)

The first 5 terms of the Binomial series for \(k=4\) are as follows

Term 1 - \(1\)

Term 2 - \(4x\)

Term 3 -\(6x^2\)

Term 4 - \(4x^3\)

Term 5 - \(x^4\)