Mister Exam

# Sum of series factorial(n)*factorial(2*n+2)/factorial(3*n)

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### The solution

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___
\  `
\   n!*(2*n + 2)!
)  -------------
/       (3*n)!
/__,
n = 1
$$\sum_{n=1}^{\infty} \frac{n! \left(2 n + 2\right)!}{\left(3 n\right)!}$$
Sum((factorial(n)*factorial(2*n + 2))/factorial(3*n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{n! \left(2 n + 2\right)!}{\left(3 n\right)!}$$
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
$$a_{n} = \frac{n! \left(2 n + 2\right)!}{\left(3 n\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{n! \left(2 n + 2\right)! \left(3 n + 3\right)!}{\left(3 n\right)! \left(n + 1\right)! \left(2 n + 4\right)!}}\right|$$
Let's take the limit
we find
False

False
The rate of convergence of the power series
$$4 {{}_{3}F_{2}\left(\begin{matrix} 1, \frac{5}{2}, 3 \\ \frac{4}{3}, \frac{5}{3} \end{matrix}\middle| {\frac{4}{27}} \right)}$$