Mister Exam

# Sum of series 8^(n-1)/factorial(n-1)

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### The solution

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  oo
____
\
\      n - 1
\    8
/   --------
/    (n - 1)!
/___,
n = 1         
$$\sum_{n=1}^{\infty} \frac{8^{n - 1}}{\left(n - 1\right)!}$$
Sum(8^(n - 1)/factorial(n - 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{8^{n - 1}}{\left(n - 1\right)!}$$
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
$$a_{n} = \frac{8^{n - 1}}{\left(n - 1\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(8^{- n} 8^{n - 1} \left|{\frac{n!}{\left(n - 1\right)!}}\right|\right)$$
Let's take the limit
we find
False

False
The rate of convergence of the power series
 8
e 
$$e^{8}$$
exp(8)
2980.957987041728274743592099
2980.957987041728274743592099`