# How do you use the binomial series to expand sqrt(1+x)?

Dec 18, 2015

sqrt(1+x) = (1+x)^(1/2) = sum(1//2)_k/(k!)x^k with $x \in \mathbb{C}$
Use the generalization of the binomial formula to complex numbers.

#### Explanation:

There is a generalization of the binomial formula to the complex numbers.

The general binomial series formula seems to be (1+z)^r = sum((r)_k)/(k!)z^k with ${\left(r\right)}_{k} = r \left(r - 1\right) \left(r - 2\right) \ldots \left(r - k + 1\right)$ (according to Wikipedia). Let's apply it to your expression.

This is a power series so obviously, if we want to have chances that this doesn't diverge we need to set $\left\mid x \right\mid < 1$ and this is how you expand $\sqrt{1 + x}$ with the binomial series.

I'm not going to demonstrate the formula is true, but it's not too hard, you just have to see that the complex function defined by ${\left(1 + z\right)}^{r}$ is holomorphic on the unit disc, calculate every derivative of it at 0, and this will give you the Taylor formula of the function, which means you can develop it as a power series on the unit disc because $\left\mid z \right\mid < 1$, hence the result.