Do not use Taylor series. Instead define and show that on now argue that it is posititive.
Prove that
using the taylor series for all i can use the taylor series for its obvious that the inequality is true for positive values.
then using the fact that works for all x less than -1.
but i am jut a bit stuck on proving the inequality for values of x
Read G. Pólya's proof of the Inequality of arithmetic and geometric means - Wikipedia, the free encyclopedia
it is interesting and involves the inequality