Mister Exam

# Sum of series factorial(n)/(n^n-1)

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### The solution

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

False

False
The rate of convergence of the power series
Sum(factorial(n)/(n^n - 1), (n, 1, oo))
Sum(factorial(n)/(n^n - 1), (n, 1, oo))`