Evaluating a sum (alternating binomial series with odd denominators)

Question

How do I evaluate the following sum (for some positive integer $m$)?: $$S=\sum_{k=0}^{m}{{m \choose k}\frac{(-1)^k}{2k+1}}$$ After expanding it looks like: $$S={m \choose 0}-\frac{1}{3}{m \choose 1} + \frac{1}{5}{m \choose 2} - \frac{1}{7}{m \choose 3}+-\dots$$ Can someone help me out?

Answer 1

This sum has the integral representation $$\sum_{k=0}^m(-1)^n\binom mk\int_0^1 x^{2k}\,dx =\int_0^1(1-x^2)^m\,dx.$$ Letting $y=x^2$ gives $$\frac12\int_0^1(1-y)^m \frac{dy}{\sqrt y}$$ which can be evaluated in terms of the beta function.