Given number:
$$\frac{\pi \frac{1}{3^{2 n + 1}}}{\left(2 n + 1\right)!}$$
It is a series of species
$$a_{n} \left(c x - x_{0}\right)^{d n}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}$$
In this case
$$a_{n} = \frac{3^{- 2 n - 1} \pi}{\left(2 n + 1\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(3^{- 2 n - 1} \cdot 3^{2 n + 3} \left|{\frac{\left(2 n + 3\right)!}{\left(2 n + 1\right)!}}\right|\right)$$
Let's take the limitwe find
False
False