= Square-root binomial series at its endpoints
{title2=$\sqrt{1-u}=1-\sum_{n\geq1}b_nu^n,\quad -1\leq u\leq1$}
Write $a_n=\binom{2n}{n}/4^n$, $a_0=1$. The <binomial series> for $\sqrt{1-u}$ has positive numbers $b_n=a_{n-1}-a_n=\binom{2n}{n}/[4^n(2n-1)]$ as the magnitudes of its nonconstant coefficients. The product $a_n=\prod_{j=1}^n[1-1/(2j)]$ tends to zero, since $\ln a_n\leq-\tfrac12\sum_{j=1}^n1/j\to-\infty$. Thus $\sum b_n=1$ by telescoping. The <Weierstrass M-test> gives <uniform convergence> and <absolute convergence> on $[-1,1]$. Continuity extends the interior identity to both endpoints. Its <radius of convergence> is one; outside this interval its terms do not tend to zero.
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