Solving exercises 25-30 at the end of 8.8 of the calculus book:
I’ll use the general formula for the binomial series expansion of \((1 + x)^k\):
\[ (1 + x)^k = \sum_{n=0}^\infty \binom{k}{n} x^n \]
where \(\binom{k}{n}\) is the binomial coefficient, calculated as:
\[ \binom{k}{n} = \frac{k(k-1)(k-2)\ldots(k-n+1)}{n!} \]
I’ll compute the first 5 terms (\(n = 0\) to \(n = 4\)) for each \(k\)-value.
The first five terms for \(k = \frac{1}{2}\) are:
\[ (1 + x)^{\frac{1}{2}} \approx 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 - \frac{5}{128}x^4 + \cdots \]
First five terms for \(k = -\frac{1}{2}\) are:
\[ (1 + x)^{-\frac{1}{2}} \approx 1 - \frac{1}{2}x + \frac{3}{8}x^2 - \frac{5}{16}x^3 + \frac{35}{128}x^4 + \cdots \]
First five terms for \(k = \frac{1}{3}\) are:
\[ (1 + x)^{\frac{1}{3}} \approx 1 + \frac{1}{3}x - \frac{1}{9}x^2 + \frac{5}{81}x^3 - \frac{10}{243}x^4 + \cdots \]
First five terms for \(k = 4\) are:
\[ (1 + x)^{4} \approx 1 + 4x + 6x^2 + 4x^3 + x^4 + \cdots \]