Normalized central binomial coefficients ($c_n=4^{-n}\left(\genfrac{}{}{0ex}{}{2n}{n}\right)$)

Set $c_n=4^{-n}\left(\genfrac{}{}{0ex}{}{2n}{n}\right)$, with $c_0=1$. Cancellation of factorials givesConsequently $c_n$ decreases to zero: $\log c_n\le -\frac{1}{2}{\sum }_{j=1}^{n}1/j\le -\frac{1}{2}\log (n+1)$. The inequality uses $\log (1-x)\le -x$ and comparison with ${\int }_1^{n+1}dt/t$. Conversely $c_n\ge 1/(n+1)$ by induction, since its ratio is at least $(n+1)/(n+2)$. Thus $\sum c_n$ diverges while $\sum (-1{)}^n{c}_n$ has conditional convergence by the alternating series test. These elementary bounds require no asymptotic formula for factorials.