What is the range of convergence of $\sum_{n=0}^{\infty} {(-1)}^n\binom{1/2}{n}\frac{1}{2n+3}.$ I was fiddling with the integral
$$\int_0^1 x^2\sqrt{1-x^2} \ dx $$
and I expanded the term under square root using a binomial series. Integrating, I got the result
$$\sum_{n=0}^{\infty} {(-1)}^n\binom{1/2}{n}\frac{x^{2n+3}}{2n+3}\Biggr|_0^1.$$

I would like to know if evaluating this series at the upper limit 1 would make it converge, since binomial series has a convergence of $|x|&lt;1$? Also if it does converge what is the range of convergence?
Since the coefficient of $x^{2n+3}$ is asymptotic to a constant times $n^{-3/2}$, and $\sum_n n^{-3/2}$ converges, this does converge for $|x| \le 1$.  The answer, btw, is $\pi/16$.
From $\int_0^1 x^2\sqrt{1-x^2} \ dx$ 

Using the following substitution: 

$t=x^2, \frac {dx}{dt}=\frac{1}{2}\frac{1}{\sqrt(t)}$ , at $x=0 \rightarrow t=0$ and at $x=1 \rightarrow t=1$

Applying the definition and identity of $\beta$ function:

$\frac{1}{2}\int\limits_0^1 t^\frac{1}{2}(1-t)^\frac{1}{2}dt= \frac{1}{2} \beta(\frac{3}{2},\frac{3}{2})=\frac{1}{2}\frac{\Gamma(\frac{3}{2})^2}{\Gamma(3)}= \frac{\pi}{16}$