Prove an infinite series identity

Prove that $1-\frac{1}{2}\cdot \frac{1}{2}+\frac{1\cdot 3}{2\cdot 4}\cdot \frac{1}{2^2}-\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\cdot \frac{1}{2^3}+\frac{1\cdot 3\cdot 5\cdot 7}{2\cdot 4\cdot 6\cdot 8}\cdot \frac{1}{2^4}-\cdots =\sqrt{\frac{2}{3}}$.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XIV. Binomial Theorem: Any Index, Examples XIV.c, No. 5