Mister Exam

# Sum of series factorial(n)/(n^3+n^2+4)

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### The solution

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

False`
The rate of convergence of the power series