= Solution
The usual interior condition for the <binomial series> is $|3x|<1$, giving \b[$-1/3<x<1/3$] as its open interval of convergence. In this particular square-root case the full series also converges absolutely at both endpoints, so the maximal real range on which its sum equals the original <function> is
$$
\boxed{-\frac13\leq x\leq\frac13.}
$$
For an endpoint check, write $\sqrt{1-u}=1-\sum_{n\geq1}b_nu^n$, where $b_n=a_{n-1}-a_n>0$ and $a_n=\binom{2n}{n}/4^n\to0$. Then $\sum b_n=1$ by telescoping, establishing <absolute convergence> at $u=\pm1$. The <square-root binomial series at its endpoints> proves <continuity> of the sum there. The endpoint values are $\sqrt2$ at $x=-1/3$ and $0$ at $x=1/3$. Outside this range the power-series terms do not tend to zero. A three-term truncation remains an approximation, rather than an exact formula throughout the interval.
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