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

1 Answer
Dec 18, 2015

Answer:

#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.