= Normalized central binomial coefficients
{title2=$c_n=4^{-n}\binom{2n}{n}$}
Set $c_n=4^{-n}\binom{2n}{n}$, with $c_0=1$. Cancellation of factorials gives
$$
c_n=\prod_{j=1}^n\left(1-\frac1{2j}\right),\qquad
\frac{c_{n+1}}{c_n}=1-\frac1{2n+2}.
$$
Consequently $c_n$ decreases to zero: $\log c_n\leq-\frac12\sum_{j=1}^n1/j\leq-\frac12\log(n+1)$. The inequality uses $\log(1-x)\leq-x$ and comparison with $\int_1^{n+1}dt/t$. Conversely $c_n\geq1/(n+1)$ by induction, since its ratio is at least $(n+1)/(n+2)$. Thus $\sum c_n$ diverges while $\sum(-1)^nc_n$ has <conditional convergence> by the <alternating series test>. These elementary bounds require no asymptotic formula for factorials.
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