Summation of $\big(\frac{1}{3}\big)^{2k+1}. \binom{2k}{k} + \dots $

Question

Summation of $\frac{1}{3} + \big(\frac{1}{3}\big)^3. \binom{2}{1} + \big(\frac{1}{3}\big)^5. \binom{4}{2} + \dots + \big(\frac{1}{3}\big)^{2k+1}. \binom{2k}{k} + \dots =$?

Answer 1

HINT: The generating function for the sequence of central binomial coefficients $\binom{2n}n$ is

$$\frac1{\sqrt{1-4x}}=\sum_{n\ge 0}\binom{2n}nx^n\;.$$

Your series is

$$\frac13\sum_{n\ge 0}\binom{2n}n\left(\frac19\right)^n\;.$$

Answer 2

$$\sum_{k=0}^{\infty }\binom{2k}kx^{2k}\;.=\frac{1}{\sqrt{1-4x^2}} \quad{x<\frac{1}{2}}$$ multiply by $x$ $$\sum_{k=0}^{\infty }\binom{2k}kx^{2k+1}=\frac{x}{\sqrt{1-4x^2}}$$