Mister Exam

# Sum of series factorial(k)

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### The solution

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  oo
__
\
)   k!
/_,
k = 0   
$$\sum_{k=0}^{\infty} k!$$
Sum(factorial(k), (k, 0, oo))
The radius of convergence of the power series
Given number:
$$k!$$
It is a series of species
$$a_{k} \left(c x - x_{0}\right)^{d k}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{k \to \infty} \left|{\frac{a_{k}}{a_{k + 1}}}\right|}{c}$$
In this case
$$a_{k} = k!$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{k \to \infty} \left|{\frac{k!}{\left(k + 1\right)!}}\right|$$
Let's take the limit
we find
False

False`
The rate of convergence of the power series