Magnetic field and taylor series

Question

I'm really confused about this, I have the magnetic field generated by a current in a disk. $$B=\frac{\mu \sigma \omega}{2}\left(\frac{\sqrt{R^2+Z^2}}{Z}+\frac{Z}{\sqrt{R^2+Z^2}}-2\right)$$ So I was asked to see what happens when Z is really big. I took the taylor series expansion of both terms separatedly, taking $\frac{R^2}{Z^2}=x$. if I do the first two $$1+\frac{1}{2}x+1-\frac{1}{2}x-2=0$$ and if i do the first three $$1+\frac{1}{2}x-\frac{1}{8}x^2+1-\frac{1}{2}x+\frac{3}{8}x^2-2=\frac{1}{4}x^2$$ and if I do the first four$$1+\frac{1}{2}x-\frac{1}{8}x^2+\frac{1}{16}x^3+1-\frac{1}{2}x+\frac{3}{8}x^2-\frac{5}{16}x^3-2=\frac{1}{4}x^2-\frac{1}{4}x^3$$ i get something negative, why? why does adding more terms "change" the result, or why do some "work" and others don't? thanks

Answer 1

I really don't see why you are troubled. There is no reason why all terms in a Taylor series should be positive, especially if you have terms like $$ \frac{Z}{\sqrt{Z^2+R^2}}=\frac{1}{\sqrt{1+R^2/Z^2}} = (1+R^2/Z^2)^{-1/2} $$ where you are expanding a negative power of a small argument $x=R^2/Z^2$. The term $-x^3/4$ simply indicates that your first approximation $x^2/4$ slightly overshoots the actual answer.

Since $R^2/Z^2<<1$, this does not affect the direction of the field but simply slightly adjusts its magnitude by a small amount.