How to find the maclaurin series for $(4+x^2)^{1/2}$

Question

The question asks to use the binomial theorem to find the macluarin series for $(4+x^2)^{1/2}$. I wanted to use the macluarin series for $\sqrt{1+x}$ which I understand how to find. Is that acceptable? I know that if you have a definition for some $f(x)$ then you also have a definition for $f(x^2)$ However, since we create this function by differentiating a certain value over and over again, it seems weird that I can just simply change the input and get the new function...is this is asilly question?

Answer 1

Use Binomial series

$(4+x^2)^{1/2}=4^{1/2}\left(1+\dfrac{x^2}4\right)^{1/2}$ for $\left|\dfrac{x^2}4\right|<1$

and $(4+x^2)^{1/2}=|x|\left(1+\dfrac4{x^2}\right)^{1/2}$for $\left|\dfrac4{x^2}\right|<1$

Answer 2

Yes, you can use substitution to find Taylor series of composite functions.

There is a theorem which proves the uniqueness of Taylor series.

That means as long as you get Taylor series for your function, that is the only Taylor series that you can get regardless of the method that you use.

For example you may find the Taylor series for $\sin 2x$ either by multiplying the two series for $ 2\sin x$ and $\cos x$ or direct substitution of $2x$ for $x$ in the Taylor series of $\sin x$