Mister Exam

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

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### The solution

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

False

False
The rate of convergence of the power series
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\  
\    -n
/   n  *n!
/__,
n = 1       
$$\sum_{n=1}^{\infty} n^{- n} n!$$
Sum(n^(-n)*factorial(n), (n, 1, oo))
1.87985386217525853348630614507
1.87985386217525853348630614507