What is the radius of convergence? $\displaystyle\sum_{n=1}^{\infty}x^{2n-1}/a_{n}$
What is the radius of convergence?
Ps: I found that $\limsup|1/a_{n}|^{1/n}=1/6$
But I am confused because of $x^{2n-1}$
What is the radius of convergence? 

Your sum should start at $n=1$. Otherwise, this is not a power series.
Then the radius of convergence of your series is the same as the one of 
$$
\sum\frac{x^{2n}}{a_n}=\sum\frac{y^n}{a_n}
$$
with $y=x^2$.
If you found $\limsup |1/a_n|^{1/n}=1/6$, this means that the $y$ power series has radius $6$.
Therefore, the $x$ power series has radius $\sqrt{6}$.
