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I would like to pose a question about the range of validity of the expansion of Binomial Theorems.

I know that for non-positive integer, rational $n$ $$ \left(1+x\right)^{n}=1+nx+\frac{n\left(n-1\right)}{2!}x^{2}+\dots, $$ where the range of validity is $\left|x\right|<1$.

My question is that if we tried to expand $\left(1+f(x)\right)^n$, where $f(x)$ is any arbitrary function defined on the reals, does it follow that we could just say that the range of validity of this expansion is just $\left|f(x)\right|<1$?

For example, could I say that the range of validity of the Binomial Theorem expansion of $\left(1+(x+2x^3)\right)^n$ is just the values of $x$ that satisfies $\left|x+2x^3\right|<1$? Or is it not as straightforward as doing such substitution?

Thanks in advance for your inputs.

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I can't understand what do you mean by range of validity, do you mean Binomial series you gave for (1+x)^n is not valid if x = 2? – Ram Jan 17 '13 at 4:45
I am considering the binomial expansion of non-positive integer, rational powers. If I understand it correctly, the infinite series converge only when $|x|<1$, and thus, it would not be valid if $x=2$. – Beer Jan 17 '13 at 5:00
okay now I understood, so you are considering $ n \in \mathbb Q - \mathbb Z^+$ – Ram Jan 17 '13 at 5:07

Yes, it really is that straightforward.

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Thank you sir. Is this reasoning extendable to any power series? Again, to illustrate with an example. I know that $$\log\left(1+x\right)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}+\dots$$ for $|x|<1$. Therefore, if I am to consider the range of validity of $$\log\left(1+\sin x\right)$$, it would be $$|\sin{x}|<1$$? – Beer Jan 17 '13 at 4:47
Yes.${}{}{}{}{}$ – Gerry Myerson Jan 17 '13 at 5:19
Thank you. I really appreciate the answers. – Beer Jan 17 '13 at 5:35
More generally, if $P(x)$ is any property of $x \in X$ that is true when $x \in S$ and $f$ is any function with values in $X$, $P(f(t))$ is true when $f(t) \in S$. – Robert Israel Jan 17 '13 at 19:00

Sorry to resurrect this post again. But I was following the suggestion above to find the range of validity of $f(x)=\log(1+\sin(x))$, and I obtained that $|\sin(x)|<1 \implies |x|<\frac{\pi}{2} \text{ or } |x-2\pi|<\frac{\pi}{2}$ etc.

When I tried to plot $f(x)$ vs its series expansion with Mathematica, I have the image found at ( Sorry, I couldn't hotlink the picture as I do not have enough rep.)

Wouldn't this imply that the answer obtained earlier only works for $|x|<\frac{\pi}{2}$, and not for other intervals like $|x-2\pi|<\frac{\pi}{2}$? How should I argue for the invalidity of the other intervals?

On another note, is range of validity of a power series the same as its radius of convergence?

Thanks again.

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