Mister Exam

# Sum of series 9^n*factorial(n)/n^(n*2)

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### The solution

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____
\
\     n
\   9 *n!
)  -----
/     n*2
/     n
/___,
n = 1      
$$\sum_{n=1}^{\infty} \frac{9^{n} n!}{n^{2 n}}$$
Sum((9^n*factorial(n))/n^(n*2), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{9^{n} n!}{n^{2 n}}$$
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
$$a_{n} = n^{- 2 n} n!$$
and
$$x_{0} = -9$$
,
$$d = 1$$
,
$$c = 0$$
then
$$R = \tilde{\infty} \left(-9 + \lim_{n \to \infty}\left(n^{- 2 n} \left(n + 1\right)^{2 n + 2} \left|{\frac{n!}{\left(n + 1\right)!}}\right|\right)\right)$$
Let's take the limit
we find
$$R = \infty$$
The rate of convergence of the power series
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___
\  
\    n  -2*n
/   9 *n    *n!
/__,
n = 1            
$$\sum_{n=1}^{\infty} 9^{n} n^{- 2 n} n!$$
Sum(9^n*n^(-2*n)*factorial(n), (n, 1, oo))
28.4718751531196540595858172056
28.4718751531196540595858172056