write down the first four terms of the following expansion (1+x)^1/4 stating in each case, the values of x for which the expansions are valid

Answer 1

To avoid writing all the powers of 1, let's agree that they are all just 1. So,

(1+x)^(1/4)
= 1 + (1/4)x^1 + (1/4)(-3/4)/2! x^2 + (1/4)(-3/4)(-7/4)/3! x^3 + ...
= 1 + x/4 - 3x^2/32 + 7x^3/128 - ...