Question

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

Answer 1

`sqrt(1+x) = (1+x)^(1/2) = sum(1//2)_k/(k!)x^k` with `x in CC`
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 `(r)_k = r(r-1)(r-2)...(r-k+1)` (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 `absx < 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 `(1+z)^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 `absz < 1`, hence the result.